Deck 8: Hypothesis Testing

Question 1

Assume that a simple random sample has been selected from a normally distributed population. Find the test statistic, P-value, critical value(s), and state the final conclusion.

-Test the claim that for the population of female college students, the mean weight is given by $\mu = 132 \mathrm { lb }$ . Sample data are summarized as $\mathrm { n } = 20 , \overline { \mathrm { x } } = 137 \mathrm { lb }$ , and $\mathrm { s } = 14.2 \mathrm { lb }$ . Use a significance level of $\alpha = 0.1$

Answer 1

$\alpha = 0.1$
Test statistic: $\mathrm { t } = 1.57$
P-value: $p = 0.1318$
Critical values: $\mathrm { t } = \pm 1.729$
Because the test statistic, $\mathrm { t } < 1.729$ , we fail to reject the null hypothesis. There is not sufficient evidence to warrant rejection of the claim that $\mu = 132 \mathrm { lb }$ .

Question 2

Jenny is testing a claim about a population mean. The hypotheses are as follows.
$\mathrm { H } _ { 0 } : \mu = 50$
$\mathrm { H } _ { 1 } : \mu > 50$
She selects a simple random sample and finds that the sample mean is 54.2. She then does some calculations and is able to make the following statement: If H0 were true, the chance that the sample mean would have come out as big ( or bigger) than 54.2 is 0.3. What name is given to the value 0.3? Do you think that she should reject the null hypothesis? Why or why not?

Answer 2

$0.3$ is the P-value. Since the P-value is large, she should not reject the null hypothesis. If $\mathrm { H } _ { 0 }$ were true, the sample mean could easily be as big as $54.2$ by chance. So there is not sufficient evidence to reject $\mathrm { H } _ { 0 }$ in favor of the alternative $\mathrm { H } _ { 1 }$ : $\mu > 50$

Question 3

Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and finalconclusion that addresses the original claim.


-A poll of 1068 adult Americans reveals that 48% of the voters surveyed prefer the democratic candidate for the presidency. At the 0.05 level of significance, test the claim that at least half of all voters prefer the Democrat.

Answer 3

$\mathrm { H } _ { 0 } : \mathrm { p } = 0.5 . \mathrm { H } _ { 1 } : \mathrm { p } < 0.5$ . Test statistic: $\mathrm { z } = - 1.31 . \mathrm { P }$ -value: $\mathrm { p } = 0.0951$
Critical value: $\mathrm { z } = - 1.645$ . Fail to reject null hypothesis. There is not sufficient evidence to warrant rejection of the claim that at least half of all voters prefer the Democrat.

Question 4

Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected.

-In one town, monthly incomes for men with college degrees are found to have a standard deviation of $650. Use a 0.01 significance level to test the claim that for men without college degrees in that town, incomes have a higher standard deviation. A random sample of 22 men without college degrees resulted in incomes with a standard deviation of $923.

Question 5

Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected.

-A manufacturer uses a new production method to produce steel rods. A random sample of 17 steel rods resulted in lengths with a standard deviation of 2.1 cm. At the 0.10 significance level, test the claim that the new production method has lengths with a standard deviation different from 3.5 cm, which was the standard deviation for the old method.

Question 6

Provide an appropriate response

-Kate asked her female friends whether they were vegetarian. Among 40 responses, 15 were responses of "yes". Is it valid to use these results to test the claim that the proportion of American women that are vegetarians is greater than 25%?

Question 7

Assume that a simple random sample has been selected from a normally distributed population and test the given claim.
Use either the traditional method or P-value method as indicated. Identify the null and alternative hypotheses, test statistic, critical value(s) or P-value (or range of P-values) as appropriate, and state the final conclusion that addresses the original claim.

-In tests of a computer component, it is found that the mean time between failures is 520 hours. A modification is made which is supposed to increase the time between failures.
Tests on a random sample of 10 modified components resulted in the following times (in hours) between failures.
$$\begin{array} { l l l l l } 518 & 548 & 561 & 523 & 536 \\499 & 538 & 557 & 528 & 563\end{array}$$
At the 0.05 significance level, test the claim that for the modified components, the mean time between failures is greater than 520 hours. Use the P-value method of testing hypotheses.

Question 8

Provide an appropriate response

-Sam wanted to test a claim about the mean of a population whose standard deviation was unknown. He picked a simple random sample of size 20 from the population. Lou wanted to test a claim about a mean of a different population whose standard deviation was known. He picked a simple random sample of size 22 from that population. George said that Sam would need to determine whether his sample was from a normally distributed population because the population standard deviation was unknown. He said that Lou would not need to do this since for his test the population standard deviation was known. Is George right?

Question 9

Assume that a simple random sample has been selected from a normally distributed population and test the given claim.
Use either the traditional method or P-value method as indicated. Identify the null and alternative hypotheses, test statistic, critical value(s) or P-value (or range of P-values) as appropriate, and state the final conclusion that addresses the original claim.

-A public bus company official claims that the mean waiting time for bus number 14 during peak hours is less than 10 minutes. Karen took bus number 14 during peak hours on 18 different occasions. Her mean waiting time was 7.3 minutes with a standard deviation of 1.5 minutes. At the 0.01 significance level, test the claim that the mean waiting time is less than 10 minutes. Use the P-value method of testing hypotheses.

Question 10

Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim

-A supplier of digital memory cards claims that no more than 1% of the cards are defective.
In a random sample of 600 memory cards, it is found that 3% are defective, but the supplier claims that this is only a sample fluctuation. At the 0.01 level of significance, test the supplier's claim that no more than 1% are defective.

Question 11

Provide an appropriate response

-In a population, 11% of people are left handed. In a simple random sample of 160 people
selected from this population, the proportion of left handers is 0.10. What is the number of
left handers in the sample and what notation is given to that number? What are the values
of p and $\hat p$ ?

Question 12

Assume that a simple random sample has been selected from a normally distributed population and test the given claim.
Use either the traditional method or P-value method as indicated. Identify the null and alternative hypotheses, test statistic, critical value(s) or P-value (or range of P-values) as appropriate, and state the final conclusion that addresses the original claim.

-A cereal company claims that the mean weight of the cereal in its packets is 14 oz. The weights (in ounces) of the cereal in a random sample of 8 of its cereal packets are listed below.
$$\begin{array} { l l l l l l l l } 14.6 & 13.8 & 14.1 & 13.7 & 14.0 & 14.4 & 13.6 & 14.2\end{array}$$ Test the claim at the 0.01 significance level.

Question 13

Assume that a simple random sample has been selected from a normally distributed population and test the given claim.
Use either the traditional method or P-value method as indicated. Identify the null and alternative hypotheses, test statistic, critical value(s) or P-value (or range of P-values) as appropriate, and state the final conclusion that addresses the original claim.

-David wants to test a claim about a population mean. The population standard deviation is unknown, the sample is a simple random sample of size 20, and the population is normally distributed. In this case, the t-test should be used since ? is unknown. If David incorrectly uses the normal distribution instead of the t-distribution, will he obtain a P-value that is too big or too small? Explain your thinking. Will he be more likely or less likely to reject the null hypothesis than if had correctly used the t-distribution?

Question 14

Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected.

-List three phrases which are associated with one-tailed claims.

Question 15

Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected.

-A machine dispenses a liquid drug into bottles in such a way that the standard deviation of the contents is 81 milliliters. A new machine is tested on a sample of 24 containers and the standard deviation for this sample group is found to be 26 milliliters. At the 0.05 level of significance, test the claim that the amounts dispensed by the new machine have a smaller standard deviation.

Question 16

Test the given claim. Use the P-value method or the traditional method as indicated. Identify the null hypothesis, alternative hypothesis, test statistic, critical value(s) or P-value, conclusion about the null hypothesis, and final
conclusion that addresses the original claim.

-A simple random sample of 15-year old boys from one city is obtained and their weights (in pounds) are listed below. Use a 0.01 significance level to test the claim that these sample weights come from a population with a mean equal to 150 lb. Assume that the standard deviation of the weights of all 15-year old boys in the city is known to be 16.4 lb. Use the traditional method of testing hypotheses. $$\begin{array} { l l l l l l l l l l l } 144 & 140 & 161 & 151 & 134 & 189 & 157 & 144 & 175 & 127 & 164\end{array}$$

Question 17

Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected.

-Systolic blood pressure levels for men have a standard deviation of 19.7 mm Hg. A random sample of 31 women resulted in blood pressure levels with a standard deviation of 23.7 mm Hg. Use a 0.05 significance level to test the claim that blood pressure levels for women have the same variation as those for men.

Question 18

Assume that a simple random sample has been selected from a normally distributed population. Find the test statistic, P-value, critical value(s), and state the final conclusion.

-Test the claim that for the population of history exams, the mean score is 80 . Sample data are summarized as $\mathrm { n } = 16 , \overline { \mathrm { x } } = 84.5$ , and $\mathrm { s } = 11.2$ . Use a significance level of $\alpha = 0.01$ .

Question 19

Assume that a simple random sample has been selected from a normally distributed population and test the given claim.
Use either the traditional method or P-value method as indicated. Identify the null and alternative hypotheses, test statistic, critical value(s) or P-value (or range of P-values) as appropriate, and state the final conclusion that addresses the original claim.

-A test of sobriety involves measuring the subject's motor skills. Twenty randomly selected sober subjects take the test and produce a mean score of 41.0 with a standard deviation of 3.7. At the 0.01 level of significance, test the claim that the true mean score for all sober subjects is equal to 35.0. Use the traditional method of testing hypotheses.

Question 20

Provide an appropriate response

-Tim believes that a coin is coming up tails less than $50 \%$ of the time. He tests the claim $p <$ $0.5$ . In 100 tosses, the coin comes up tails 57 times. What is the value of the sample proportion? Do you think the P-value will be small or large and what should Tim conclude about the claim $p < 0.5$ ?

Question 21

Assume that a simple random sample has been selected from a normally distributed population and test the given claim.
Use either the traditional method or P-value method as indicated. Identify the null and alternative hypotheses, test statistic, critical value(s) or P-value (or range of P-values) as appropriate, and state the final conclusion that addresses the original claim.

-A researcher wants to test the claim that convicted burglars spend an average of $18.7$ months in jail. She takes a random sample of 11 such cases from court files and finds that $\bar { x }$ $= 21.3$ months and $\mathrm { s } = 7.7$ months. Test the claim that $\mu = 18.7$ months at the $0.05$ significance level. Use the traditional method of testing hypotheses.

Question 22

Provide an appropriate response.


-Complete the following table on hypothesis testing. $$\begin{array} { c | c | c } \text { Test about } & \text { Distribution } & \text { Assumptions } \\\hline \text { Mean } & & \\\text { Mean } & & \\\text { Proportion } & & \\\text { Variance } & &\end{array}$$

Question 23

Explain how to determine if a hypothesis test is one-tailed or two-tailed and explain how you know where to shade the critical region. Give an example for each which includes the claim, the hypotheses, and the diagram with the critical region shaded.

Question 24

What do you conclude about the claim below? Do not use formal procedures or exact calculations. Use only the rare event rule and make a subjective estimate to determine whether the event is likely.

Claim: An employee of a company is equally likely to take a sick day on any day of the week. Last year, the total number of sick days taken by all the employees of the company was 143. Of these, 52 were Mondays, 14 were Tuesdays, 17 were Wednesdays, 17 were Thursdays, and 43 were Fridays.

Question 25

Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected.

-At the $\alpha = 0.05$ significance level test the claim that a population has a standard deviation of 20.3. A random sample of 18 people yields a standard deviation of 27.1.

Question 26

Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim.

-Various temperature measurements are recorded at different times for a particular city.
The mean of 20°C is obtained for 60 temperatures on 60 different days. Assuming that $\sigma = 1.5 ^ { \circ } \mathrm { C }$ test the claim that the population mean is 22°C. Use a 0.05 significance level.

Question 27

Assume that a simple random sample has been selected from a normally distributed population and test the given claim.
Use either the traditional method or P-value method as indicated. Identify the null and alternative hypotheses, test statistic, critical value(s) or P-value (or range of P-values) as appropriate, and state the final conclusion that addresses the original claim.

-A large software company gives job applicants a test of programming ability and the mean for that test has been 160 in the past. Twenty-five job applicants are randomly selected from one large university and they produce a mean score and standard deviation of 183 and 12, respectively. Use a 0.05 level of significance to test the claim that this sample comes from a population with a mean score greater than 160. Use the P-value method of testing hypotheses.

Question 28

Suppose the claim is in the alternate hypothesis. What form does your conclusion take?
Suppose the claim is in the null hypothesis. What form does your conclusion take?

Question 29

Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim.

-A nationwide study of American homeowners revealed that 64% have one or more lawn mowers. A lawn equipment manufacturer, located in Omaha, feels the estimate is too low for households in Omaha. Can the value 0.64 be rejected if a survey of 490 homes in Omaha yields 331 with one or more lawn mowers? Use $\text { Use } \alpha = 0.05$

Question 30

Assume that a simple random sample has been selected from a normally distributed population and test the given claim.
Use either the traditional method or P-value method as indicated. Identify the null and alternative hypotheses, test statistic, critical value(s) or P-value (or range of P-values) as appropriate, and state the final conclusion that addresses the original claim.

-A light-bulb manufacturer advertises that the average life for its light bulbs is 900 hours. A random sample of 15 of its light bulbs resulted in the following lives in hours.
$$\begin{array} { l l l l l l l l } 995 & 590 & 510 & 539 & 739 & 917 & 571 & 555 \\916 & 728 & 664 & 693 & 708 & 887 & 849 &\end{array}$$
At the 10% significance level, test the claim that the sample is from a population with a mean life of 900 hours. Use the P-value method of testing hypotheses.

Question 31

A manufacturer makes ball bearings that are supposed to have a mean weight of 30 g. A retailer suspects that the mean weight is actually less than 30 g. The mean weight for a random sample of 16 ball bearings is 29.2 g with a standard deviation of 4.2 g. At the 0.05 significance level, test the claim that the sample comes from a population with a mean weight less than 30 g. Use the traditional method of testing hypotheses.

Question 32

Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim

-The health of employees is monitored by periodically weighing them in. A sample of 54 employees has a mean weight of 183.9 lb. Assuming that ? is known to be 121.2 lb, use a 0.10 significance level to test the claim that the population mean of all such employees weights is less than 200 lb.

Question 33

What do you conclude about the claim below? Do not use formal procedures or exact calculations. Use only the rare event rule and make a subjective estimate to determine whether the event is likely.

Claim: A die is fair and in 100 rolls there are 63 sixes.

Question 34

Test the given claim. Use the P-value method or the traditional method as indicated. Identify the null hypothesis,
alternative hypothesis, test statistic, critical value(s) or P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim.

-The maximum acceptable level of a certain toxic chemical in vegetables has been set at 0.4 parts per million (ppm). A consumer health group measured the level of the chemical in a random sample of tomatoes obtained from one producer. The levels, in ppm, are shown below. $$\begin{array} { l l l l l } 0.31 & 0.47 & 0.19 & 0.72 & 0.56 \\0.91 & 0.29 & 0.83 & 0.49 & 0.28 \\0.31 & 0.46 & 0.25 & 0.34 & 0.17 \\0.58 & 0.19 & 0.26 & 0.47 & 0.81\end{array}$$
Do the data provide sufficient evidence to support the claim that the mean level of the chemical in tomatoes from this producer is greater than the recommended level of 0.4 ppm? Use a 0.05 significance level to test the claim that these sample levels come from a population with a mean greater than 0.4 ppm. Use the P-value method of testing hypotheses. Assume that the standard deviation of levels of the chemical in all such tomatoes is 0.21 ppm.

Question 35

Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected.

-With individual lines at the checkouts, a store manager finds that the standard deviation for the waiting times on Monday mornings is 5.7 minutes. After switching to a single waiting line, he finds that for a random sample of 29 customers, the waiting times have a standard deviation of 4.9 minutes. Use a 0.025 significance level to test the claim that with a single line, waiting times vary less than with individual lines.

Question 36

Provide an appropriate response

-Compare the steps in the traditional method of hypothesis testing with the steps in the P-value method of hypothesis testing. How are they alike and how are they different? $$\begin{array} { | l | l | l | } \hline \text { Traditional Method } & \text { P-value Method } & \text { Comparison } \\\hline & & \\\hline & & \\\hline & & \\\hline & & \\\hline & & \\\hline & & \\\hline & & \\\hline & & \\\hline & & \\\hline & & \\\hline\end{array}$$

Question 37

Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim

-A manufacturer considers his production process to be out of control when defects exceed 3%. In a random sample of 85 items, the defect rate is 5.9% but the manager claims that this is only a sample fluctuation and production is not really out of control. At the 0.01 level of significance, test the manager's claim.

Question 38

Suppose that you perform a hypothesis test regarding a population mean, and that the evidence does not warrant rejection of the null hypothesis. When formulating the conclusion to the test, why is the phrase "fail to reject the null hypothesis" more accurate than the phrase "accept the null hypothesis"?

Question 39

Discuss the rationale for hypothesis testing. Refer to the comparison of the assumption and the sample results.

Question 40

Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected.

-The standard deviation of math test scores at one high school is 16.1. A teacher claims that the standard deviation of the girls' test scores is smaller than 16.1. A random sample of 22 girls results in scores with a standard deviation of 12. Use a significance level of 0.01 to test the teacher's claim.

Question 41

Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim.

-In a sample of 167 children selected randomly from one town, it is found that 37 of them suffer from asthma. At the 0.05 significance level, test the claim that the proportion of all children in the town who suffer from asthma is 11%.

Question 42

Assume that a simple random sample has been selected from a normally distributed population and test the given claim.
Use either the traditional method or P-value method as indicated. Identify the null and alternative hypotheses, test statistic, critical value(s) or P-value (or range of P-values) as appropriate, and state the final conclusion that addresses the original claim.

-Use a significance level of $\alpha = 0.05$ to test the claim that $\mu = 32.6$ . The sample data consist of 15 scores for which $\bar { x } = 41.6$ and $s = 8$ . Use the traditional method of testing hypotheses.

Question 43

Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim

-A random sample of 100 pumpkins is obtained and the mean circumference is found to be 40.5 cm. Assuming that the population standard deviation is known to be 1.6 cm, use a 0.05 significance level to test the claim that the mean circumference of all pumpkins is equal to 39.9 cm.

Question 44

Write the claim that is suggested by the given statement, then write a conclusion about the claim. Do not use symbolic expressions or formal procedures; use common sense.

A person claims to have extra sensory powers. A card is drawn at random from a deck of cards and without looking at the card, the person is asked to identify the suit of the card.
He correctly identifies the suit 28 times out of 100.

Question 45

Assume that a simple random sample has been selected from a normally distributed population and test the given claim.
Use either the traditional method or P-value method as indicated. Identify the null and alternative hypotheses, test statistic, critical value(s) or P-value (or range of P-values) as appropriate, and state the final conclusion that addresses the original claim.

-Use a significance level of $\alpha = 0.01$ to test the claim that $\mu > 2.85$ . The sample data consist of 9 scores for which $\bar { x } = 3.19$ and $s = 0.55$ . Use the traditional method of testing hypotheses.

Question 46

Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim

-An article in a journal reports that 34% of American fathers take no responsibility for child care. A researcher claims that the figure is higher for fathers in the town of Littleton. A random sample of 234 fathers from Littleton yielded 96 who did not help with child care.
Test the researcher's claim at the 0.05 significance level.

Question 47

Write the claim that is suggested by the given statement, then write a conclusion about the claim. Do not use symbolic expressions or formal procedures; use common sense.
A math teacher tries a new method for teaching her introductory statistics class. Last year the mean score on the final test was 73. This year the mean on the same final was 76.

Question 48

Complete the table to compare the z and t distributions. $$\begin{array} { c | c | c } & z \text { distribution } & t \text { distribution } \\\hline \text { Shape } & & \\\text { Mean value } & & \\\text { Standard deviation value } & & \\\text { Requirements } & &\end{array}$$

Question 49

Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim

-In a clinical study of an allergy drug, 108 of the 202 subjects reported experiencing significant relief from their symptoms. At the 0.01 significance level, test the claim that more than half of all those using the drug experience relief.

Question 50

What do you conclude about the claim below? Do not use formal procedures or exact calculations. Use only the rare event rule and make a subjective estimate to determine whether the event is likely.

Claim: A company claims that the proportion of defective units among a particular model of computers is 4%. In a shipment of 200 such computers, there are 10 defective units.

Question 51

Suppose that you are conducting a study on the effectiveness of a new teaching method and that you wish to use a hypothesis test to support your claim regarding the mean test score under this method. What restrictions are there in the wording of the claim? Will your claim become the null hypothesis or the alternative hypothesis, or does it depend on the situation? Give an example of a claim which is incorrectly worded.

Question 52

Assume that a simple random sample has been selected from a normally distributed population. Find the test statistic, P-value, critical value(s), and state the final conclusion.

-Test the claim that the mean age of the prison population in one city is less than 26 years. Sample data are summarized as $\mathrm { n } = 25 , \overline { \mathrm { x } } = 24.4$ years, and $\mathrm { s } = 9.2$ years. Use a significance level of $\alpha = 0.05$ .

Question 53

Assume that a simple random sample has been selected from a normally distributed population. Find the test statistic, P-value, critical value(s), and state the final conclusion.

-Test the claim that for the adult population of one town, the mean annual salary is given by $\mu = \$ 30,000$ . Sample data are summarized as $n = 17 , \bar { x } = \$ 22,298$ , and $s = \$ 14,200$ . Use a significance level of $\alpha = 0.05$ .

Question 54

Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim.

-According to a recent poll 53% of Americans would vote for the incumbent president. If a random sample of 100 people results in 45% who would vote for the incumbent, test the claim that the actual percentage is 53%. Use a 0.10 significance level.

Question 55

Under what conditions do you reject $\mathrm { H } _ { 0 }$ ? Discuss both the traditional and the P-value approach.

Question 56

Suppose that you wish to use a hypothesis test to test a claim made by a juice bottling company regarding the mean amount of juice in its 16 oz bottles. Why does the original claim sometimes become the null hypothesis, and why does it sometimes become the alternative hypothesis? Give an example of a claim which would become the null hypothesis and an example of a claim would become the alternative hypothesis.

Question 57

When testing hypotheses about a mean, the decision must be made as to the distribution to be used. Discuss the decision process used to decide whether z or t or neither is the proper distribution.

Question 58

Write the claim that is suggested by the given statement, then write a conclusion about the claim. Do not use symbolic expressions or formal procedures; use common sense.
Last year an appliance manufacturer received many complaints about the high rate of defects among its washing machines. Approximately 9% of the machines were defective in some way. This year the company tightened up its quality control procedures. The latest shipment of 250 washing machines contained 2 defective units.

Question 59

What do you conclude about the claim below? Do not use formal procedures or exact calculations. Use only the rare event rule and make a subjective estimate to determine whether the event is likely.

Claim: A roulette wheel is fair and in 40 consecutive spins of the wheel, black shows up 23 times. (A roulette wheel has 38 equally likely slots of which 18 are black).

Question 60

Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected.

-When 12 bolts are tested for hardness, their indexes have a standard deviation of 41.7. Test the claim that the standard deviation of the hardness indexes for all such bolts is greater than 30.0. Use a 0.025 level of significance.

Question 61

Find the critical value or values of $\chi ^ { 2 }$ based on the given information.

- $\begin{array} { l } \mathrm { H } _ { 0 } : \sigma = 8.0 \\\mathrm { n } = 10 \\\alpha = 0.01\end{array}$

A) 1.735, 23.589
B) 2.088, 21.666
C) 21.666
D) 23.209

Question 62

Assume that a hypothesis test of the given claim will be conducted. Identify the type I or type II error for the test.

-For large numbers of degrees of freedom, the critical $\chi ^ { 2 }$ values can be approximated as follows:
$$\chi ^ { 2 } = \frac { 1 } { 2 } ( z + \sqrt { 2 k - 1 } ) ^ { 2 }$$
where $\mathrm { k }$ is the number of degrees of freedom and $\mathrm { z }$ is the critical value. To find the lower critical value, the negative z-value is used, to find the upper critical value, the positive z-value is used. Use this approximation to estimate the critical value of $\chi ^ { 2 }$ in a right-tailed hypothesis test with $\mathrm { n } = 146$ and $\alpha = 0.01$

A) $\chi ^ { 2 } \approx 192.742$
B) $\chi ^ { 2 } \approx 186.824$
C) $\chi ^ { 2 } \approx 187.961$
D) $\chi ^ { 2 } \approx 191.590$

Question 63

Write the claim that is suggested by the given statement, then write a conclusion about the claim. Do not use symbolic expressions or formal procedures; use common sense.
Of a group of 1000 people suffering from arthritis, 500 receive acupuncture treatment and 500 receive a placebo. Among those in the placebo group, 24% noticed an improvement, while of those receiving acupuncture, 44% noticed an improvement.

Question 64

Assume that a simple random sample has been selected from a normally distributed population. Find the test statistic, P-value, critical value(s), and state the final conclusion.

-Test the claim that the mean lifetime of car engines of a particular type is greater than 220,000 miles. Sample data are summarized as $n = 23 , \bar { x } = 226,450$ miles, and $\mathrm { s } = 11,500$ miles. Use a significance level of $\alpha = 0.01$

Question 65

Use the given information to find the P-value. Also, use a 0.05 significance level and state the conclusion about the null hypothesis (reject the null hypothesis or fail to reject the null hypothesis).

-The test statistic in a two-tailed test is z = 1.95.

A) 0.0256; reject the null hypothesis
B) 0.0512; fail to reject the null hypothesis
C) 0.0512; reject the null hypothesis
D) 0.9744; fail to reject the null hypothesis

Question 66

Formulate the indicated conclusion in nontechnical terms. Be sure to address the original claim.

-A psychologist claims that more than 24 percent of the population suffers from professional problems due to extreme shyness. Assuming that a hypothesis test of the claim has been conducted and that the conclusion is failure to reject the null hypothesis, state the conclusion in nontechnical terms.

A) There is not sufficient evidence to support the claim that the true proportion is greater than 24 percent.
B) There is not sufficient evidence to support the claim that the true proportion is less than 24 percent.
C) There is sufficient evidence to support the claim that the true proportion is less than 24 percent.
D) There is sufficient evidence to support the claim that the true proportion is greater than 24 percent.

Question 67

Find the value of the test statistic z using z $$z = \frac { \hat { p } - p } { \sqrt { \frac { p q } { n } } }$$

-A claim is made that the proportion of children who play sports is less than 0.5, and the sample statistics include n = 1899 subjects with 30% saying that they play a sport.

A) -17.43
B) -35.58
C) 17.43
D) 35.58

Question 68

Suppose that you wish to test a claim about a population mean. Which distribution should be used given that the sample is a voluntary response sample, $\sigma$ is unknown, n = 15, and the population is normally distributed?

A) Neither the normal nor the t-distribution
B) t-distribution
C) Normal distribution

Question 69

The mean resting pulse rate for men is 72 beats per minute. A simple random sample of men who regularly work out at Mitch's Gym is obtained and their resting pulse rates (in beats per minute) are listed below. Use a 0.05 significance level to test the claim that these sample pulse rates come from a population with a mean less than 72 beats per minute. Assume that the standard deviation of the resting pulse rates of all men who work out at
Mitch's Gym is known to be 6.8 beats per minute. Use the traditional method of testing hypotheses. $$\begin{array} { l l l l l l l } 54 & 60 & 67 & 84 & 74 & 64 & 69\\70 & 66 & 80 & 59 & 71 & 76 & 63\\\end{array}$$

Question 70

Formulate the indicated conclusion in nontechnical terms. Be sure to address the original claim.

-An entomologist writes an article in a scientific journal which claims that fewer than 17 in ten thousand male fireflies are unable to produce light due to a genetic mutation. Assuming that a hypothesis test of the claim has been conducted and that the conclusion is to reject the null hypothesis, state the conclusion in nontechnical terms.

A) There is sufficient evidence to support the claim that the true proportion is greater than 17 in ten thousand.
B) There is not sufficient evidence to support the claim that the true proportion is greater than 17 in ten thousand.
C) There is not sufficient evidence to support the claim that the true proportion is less than 17 in ten thousand.
D) There is sufficient evidence to support the claim that the true proportion is less than 17 in ten thousand.

Question 71

In a hypothesis test, which of the following will cause a decrease in $\beta$ , the probability of making a type II error?

A: Increasing $\alpha$ while keeping the sample size $n$ , fixed
B: Increasing the sample size $n$ , while keeping $\alpha$ fixed
$\mathrm { C }$ : Decreasing $\alpha$ while keeping the sample size $n$ , fixed
D: Decreasing the sample size $n$ , while keeping $\alpha$ fixed

A) A and D
B) A and B
C) C and D
D) B and C

Question 72

Assume that the data has a normal distribution and the number of observations is greater than fifty. Find the critical z value used to test a null hypothesis.

- $\alpha = 0.08 ; \mathrm { H } _ { 1 }$ is $\mu \neq 3.24$

A) $\pm 1.41$
B) $1.75$
C) $\pm 1.75$
D) $1.41$

Question 73

Define Type I and Type II errors. Give an example of a Type I error which would have serious consequences. Give an example of a Type II error which would have serious consequences. What should be done to minimize the consequences of a serious Type I error?

Question 74

Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected.

-For randomly selected adults, IQ scores are normally distributed with a standard deviation of 15. The scores of 14 randomly selected college students are listed below. Use a 0.10 significance level to test the claim that the standard deviation of IQ scores of college students is less than 15. Round the sample standard deviation to three decimal places.
$$\begin{array} { l l l l l l l } 115 & 128 & 107 & 109 & 116 & 124 & 135 \\127 & 115 & 104 & 118 & 126 & 129 & 133\end{array}$$

Question 75

Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected.

-Heights of men aged 25 to 34 have a standard deviation of 2.9. Use a 0.05 significance level to test the claim that the heights of women aged 25 to 34 have a different standard deviation. The heights (in inches) of 16 randomly selected women aged 25 to 34 are listed below. Round the sample standard deviation to five decimal places. $$\begin{array} { l l l l l l l l } 62.13 & 65.09 & 64.18 & 66.72 & 63.09 & 61.15 & 67.50 & 64.65 \\63.80 & 64.21 & 60.17 & 68.28 & 66.49 & 62.10 & 65.73 & 64.72\end{array}$$

Question 76

A manufacturer finds that in a random sample of 100 of its CD players, 96% have no defects. The manufacturer wishes to make a claim about the percentage of nondefective CD players and is prepared to exaggerate. What is the highest rate of nondefective CD players that the manufacturer
Could claim under the following condition?

His claim would not be rejected at the 0.05 significance level if this sample data were used. Assume that a left-tailed hypothesis test would be performed.

A) 98.4%
B) 96.5%
C) 98.2%
D) 96.6%

Question 77

Assume that a hypothesis test of the given claim will be conducted. Identify the type I or type II error for the test.
A skeptical paranormal researcher claims that the proportion of Americans that have seen a UFO is less than 2 in a thousand. Identify the type I error for the test.

A) Reject the claim that the proportion of Americans that have seen a UFO is equal to 2 in a thousand when that proportion is actually less than 2 in a thousand.
B) Reject the claim that the proportion of Americans that have seen a UFO is equal to 2 in a thousand when that proportion is actually 2 in a thousand.
C) Fail to reject the claim that the proportion of Americans that have seen a UFO is equal to 2 in a thousand when that proportion is actually less than 2 in a thousand.
D) Fail to reject the claim that the proportion of Americans that have seen a UFO is equal to 2 in a thousand when that proportion is actually greater than 2 in a thousand.

Question 78

Describe how to write the null and alternative hypotheses based on a claim. You may give an example to clarify your explanation.

Question 79

Use the P-value method to test the claim that the population standard deviation of the systolic blood pressures of adults aged 40-50 is equal to $22 \mathrm { mmHg }$ . The sample statistics are as follows: $\mathrm { n } = 23 , \overline { \mathrm { x } } = 132.3 \mathrm { mmHg } , \mathrm { s } = 26.9 \mathrm { mmHg }$ . Be sure to state the hypotheses, the value of this test statistic, the P-value, and your conclusion. Use a significance level of $0.05$ .

Question 80

Formulate the indicated conclusion in nontechnical terms. Be sure to address the original claim.

-The manufacturer of a refrigerator system for beer kegs produces refrigerators that are supposed to maintain a true mean temperature, µ, of 42°F, ideal for a certain type of German pilsner. The owner of the brewery does not agree with the refrigerator manufacturer, and claims he can prove that the true mean temperature is incorrect. Assuming that a hypothesis test of the claim has been conducted and that the conclusion is to reject the null hypothesis, state the conclusion in nontechnical terms.

A) There is sufficient evidence to support the claim that the mean temperature is equal to 42°F.
B) There is not sufficient evidence to support the claim that the mean temperature is equal to 42°F.
C) There is sufficient evidence to support the claim that the mean temperature is different from 42°F.
D) There is not sufficient evidence to support the claim that the mean temperature is different from 42°F.