Exam 10: Hypothesis Tests Involving a Sample Mean or Proportion

A Type II error is the probability of failing to reject a ____________________.It is represented by the Greek letter ____________________.

Golf A round of golf on your typical course requires at least 4.5 hours.A particular golf club claims that the average length of time to complete a round of golf on their course is less than 4.5 hours.To test this claim,the club sampled 15 golfers and recorded their time to finish their round.The sample average was 4.4 hours with a standard deviation of 0.8 hours.Assume the time to complete a round of golf is normally distributed and use $\alpha$ = 0.01. -Use the following information to test at a 0.01 level of significance.n = 400; p = 0.03; H₀: $\pi \geq$ 0.05,H₁: $\pi$ < 0.05. State the conclusion.

In conducting a hypothesis test about the population mean $\mu$ ,if the population is truly or approximately normally distributed with $\text { ? }$ unknown,then the test statistic will be z,corresponding to the normal distribution.

An executive with a fast food restaurant believes a street located near a major highway interchange will be a profitable location.A random sample of sixty days is taken to estimate the average cars per day passing by the location.On the basis of the sample data,the executive concludes that the average is not high enough to yield a profitable outlet.Six months later a competitor builds at the same location and reports it to be one of its best-profit new stores.Describe the type of error made by the executive.

A parents group in New York City was concerned with the alcohol content in an over-the-counter cough syrup.They felt that the content was greater than the 2 cc/bottle stated on the label.A sample of 30 bottles showed an average alcohol content of 3.1 cc/bottle with a standard deviation of 2 cc./bottle.Using a 0.05 level of significance and assuming a normal distribution,does the sample indicate that the alcoholic content is above 2 cc/bottle? Appropriate test: ____________________ Test statistic = ____________________ Conclusion: ____________________ Interpretation: _________________________________________________________

If the p-value for a particular hypothesis test is 0.03 with $\alpha$ = 0.05,we do not reject the null hypothesis.

If a test is carried out at a specified significance level (e.g., $\alpha$ = .05),using a larger sample size will change the decision rule but will not change $\alpha$ .This is because $\alpha$ has been decided upon in advance.

In a two-tail test for the population mean,if the null hypothesis is rejected when the alternative hypothesis is true then:

H₀: $\mu \leq \mu _ { 0 }$ ,H₁: $\mu > \mu _ { 0 }$ ; n > 30 and $\alpha$ = 0.01 Find $z _ { \alpha }$

Suppose that 9 observations are drawn from a normal population whose standard deviation is 2.The observations are: 15,9,13,11,8,12,11,7,and 10.At 95% confidence,you want to determine whether the mean of the population from which this sample was taken is significantly different from 10. A)State the null and alternative hypotheses. H₀: ____________________ H₁: ____________________ B)Compute the value of the test statistic. C)Determine the rejection region. D)Interpret the results.

In a one-tail test for the population mean,if the null hypothesis is not rejected when the alternative hypothesis is true then:

A portfolio manager was analyzing the price-earnings ratio for this year's performance.His boss said that the average price-earnings ratio was 20 for the many stocks,which his firm had traded,but he felt that figure was too high.The portfolio manager randomly selected a sample of 50 P/E ratios and found a mean of 18.17 and standard deviation of 4.60.Assume that the population is normally distributed and test at the 0.01 level of significance. The decision is: Because the test statistic t = -2.83 is not less than -2.33,fail to reject H0.At the 0.01 level,the sample data suggest that the average P/E ratio for the stocks is not less than 20.

Local Restaurant A local restaurant claimed in an advertisement that at least 80% of their customers rate their establishment as "Excellent".An independent survey company asked 200 customers their opinion and from this random sample,168 customers rated the establishment "Excellent".Use $\alpha$ = 0.05. -H₀: $\mu \geq \mu _ { 0 }$ ,H₁: $\mu < \mu _ { 0 }$ ; n > 30 and $\alpha$ = 0.10. State the appropriate decision rule.

A professor of math refutes the claim that the average student spends 3 hours studying for the midterm exam.Which hypothesis is used to test the claim?

Light Bulb Mnf. A light bulb manufacturer claims that the average life of their light bulb is more than 1000 hours.To test this claim,a sample of 32 light bulbs had an average life of 1120 hours.Assume the population standard deviation is 325 hours and use $\alpha$ = 0.02. -What is your conclusion?

The p-value is the lowest possible level of significance at which the alternative hypothesis can be rejected.

Light Bulb Mnf. A light bulb manufacturer claims that the average life of their light bulb is more than 1000 hours.To test this claim,a sample of 32 light bulbs had an average life of 1120 hours.Assume the population standard deviation is 325 hours and use $\alpha$ = 0.02. -State your conclusion using the p-value.

Adult drink A particular study claims that the average U.S.adult drinks 1.7 cups of coffee per day.To test this claim,a sample of 35 adults were asked about their coffee consumption.The sample average was 1.95 cups per day.Assume the population standard deviation is 0.5 cups per day and set $\alpha$ = 0.10. -State the null and alternative hypothesis.

Table 1 Based on the sample data,a confidence interval has been constructed such that we have 95% confidence that the population mean is between 75 and 105. -For a sample of 36 items from a population for which the standard deviation $\alpha$ = 12,the sample mean is 350.Construct a 95% confidence interval for the population mean. ____________________ to ____________________

A Type I error is the probability of failing to reject a false null hypothesis.