Exam 7: Inferences Based on a Single Sample: 355 Tests of Hypotheses

We have created a $99 \%$ confidence interval for $\mu$ with the result $( 10,15 )$ . What conclusion will we make if we test $H _ { 0 } : \mu = 17$ vs. $H _ { \mathrm { a } } : \mu \neq 17$ at $\alpha = .01$ ?

For the given binomial sample size and null-hypothesized value of p0, determine whether the sample size is large enough to use the normal approximation methodology to conduct a test of the null hypothesis $\mathrm { H } _ { 0 } : \mathrm { p } = \mathrm { P } 0$ - $\mathrm { n } = 80 , \mathrm { p } 0 = 0.4$

A random sample of $n$ observations, selected from a normal population, is used to test the null hypothesis $H _ { 0 } : \sigma ^ { 2 } = 155$ . Specify the appropriate rejection region. $H _ { \mathrm { a } } : \sigma ^ { 2 } < 155 , n = 14 , \alpha = .01$

For the given binomial sample size and null-hypothesized value of p₀, determine whether the sample size is large enough to use the normal approximation methodology to conduct a test of the null hypothesis H₀: p = p₀. - $\mathrm { n } = 600 , \mathrm { p } _0 = 0.01$

A new apparatus has been devised to replace the needle in administering vaccines. The apparatus, which is connected to a large supply of vaccine, can be set to inject different amounts of the serum, but the variance in the amount of serum injected to a given person must not be greater than $.07$ to ensure proper inoculation. A random sample of 25 injections was measured. Suppose the $p$ -value for the test is $p = .0031$ . State the proper conclusion using $\alpha = .01$ .

The business college computing center wants to determine the proportion of business students who have laptop computers. If the proportion differs from 25%, then the lab will modify a proposed enlargement of its facilities. Suppose a hypothesis test is conducted and the test statistic is 2.4. Find the p-value for a two-tailed test of hypothesis.

A bottling company produces bottles that hold 12 ounces of liquid. Periodically, the company gets complaints that their bottles are not holding enough liquid. To test this claim, the bottling company randomly samples 25 bottles and finds the average amount of liquid held by the bottles is 11.8 ounces with a standard deviation of .4 ounce. Which of the following is the set of hypotheses the company wishes to test?

An educational testing service designed an achievement test so that the range in student scores would be greater than 300 points. To determine whether the objective was achieved, the testing service gave the test to a random sample of 42 students and found that the sample mean and variance were 703 and 2574, respectively. Specify the null and alternative hypotheses for determining whether the test achieved the desired dispersion in scores. Assume that range $= 6 \sigma$

A recipe submitted to a magazine by one of its subscribers states that the mean baking time for a cheesecake is 55 minutes. A test kitchen preparing the recipe before it is published in the magazine makes the cheesecake 10 times at different times of the day in different ovens. The following baking times (in minutes) are observed. 54 55 58 59 59 60 61 61 62 65 Assume that the baking times belong to a normal population. Test the null hypothesis that the mean baking time is 55 minutes against the alternative hypothesis $\mu > 55$ . Use $\alpha = .05$ .

According to an advertisement, a strain of soybeans planted on soil prepared with a specified fertilizer treatment has a mean yield of 103 bushels per acre. Fifteen farmers who belong to a cooperative plant the soybeans in soil prepared as specified. Each uses a 40 -acre plot and records the mean yield per acre. The mean and variance for the sample of the 15 farms are $\bar { x } = 88$ and $s ^ { 2 } = 10,125$ . Find the rejection region used for determining if the mean yield for the soybeans is not equal to 103 bushels per acre. Use $\alpha = .05$ . Answer: The rejection region requires $\alpha / 2 = .05 / 2 = .025$ in both tails of the $t$ distribution with $\mathrm { df } = n - 1 = 15 - 1 = 14$ . The rejection region is $t > 2.145$ or $t < - 2.145$ .

A sample of 8 measurements, randomly selected from a normally distributed population, resulted in the following summary statistics: $\bar { x } = 5.2 , s = 1.1 .$ . Test the null hypothesis that the mean of the population is 4 against the alternative hypothesis $\mu \neq 4 . \text { Use } \alpha = .05$

An industrial supplier has shipped a truckload of teflon lubricant cartridges to an aerospace customer. The customer has been assured that the mean weight of these cartridges is in excess of the 10 ounces printed on each cartridge. To check this claim, a sample of $n = 10$ cartridges are randomly selected from the shipment and carefully weighed. Summary statistics for the sample are: $\bar { x } = 10.11$ ounces, $s = .30$ ounce. To determine whether the supplier's claim is true, consider the test, $H _ { 0 } : \mu = 10$ vs. $H _ { \mathrm { a } } : \mu > 10$ , where $\mu$ is the true mean weight of the cartridges. Find the rejection region for the test using $\alpha = .01$ .

Given $H _ { 0 } : \mu = 18 , H _ { \mathrm { a } } : \mu < 18$ , and $p = 0.068$ . Do you reject or fail to reject $H _ { 0 }$ at the $.05$ level of significance?

We do not accept $H _ { 0 }$ because we are concerned with making a Type II error.

It has been estimated that the $G$ -car obtains a mean of 40 miles per gallon on the highway, and the company that manufactures the car claims that it exceeds this estimate in highway driving. To support its assertion, the company randomly selects $49 \mathrm { G }$ -cars and records the mileage obtained for each car over a driving course similar to that used to obtain the estimate. The following data resulted: $\bar { x } = 41.5$ miles per gallon, $s = 7$ miles per gallon. Calculate the power of the test if the true value of the mean is 41 miles per gallon. Use a value of $\alpha = .025$ .

A company claims that 9 out of 10 doctors (i.e., $90 \%$ ) recommend its brand of cough syrup to their patients. To test this claim against the alternative that the actual proportion is less than $90 \%$ , a random sample of doctors was taken. Suppose the test statistic is $z = - 2.23$ . Can we conclude that $H _ { 0 }$ should be rejected at the a) $\alpha = .10$ , b) $\alpha = .05$ , and c) $\alpha = .01$ level?

If a hypothesis test were conducted using ? = 0.05, to which of the following p-values would cause the null hypothesis to be rejected.

A bottling company produces bottles that hold 12 ounces of liquid. Periodically, the company gets complaints that their bottles are not holding enough liquid. To test this claim, the bottling company randomly samples 64 bottles and finds the average amount of liquid held by the bottles is 11.9155 ounces with a standard deviation of 0.40 ounce. Suppose the p-value of this test is 0.0455. State the proper conclusion.

In a test of $H _ { 0 } : \mu = 70$ against $H _ { \mathrm { a } } : \mu \neq 70$ , the sample data yielded the test statistic $z = 2.11$ . Find and interpret the $p$ -value for the test.