Exam 11: Introduction to Hypothesis Testing

A sample is used to obtain a 95% confidence interval for the mean of a population.The confidence interval goes from 78.21 to 87.64.If the same sample had been used to test the null hypothesis that the mean of the population differs from 90, the null hypothesis could be rejected at a level of significance of 0.05.

For a given level of significance, if the sample size is increased, the probability of committing a Type II error will increase.

If a null hypothesis is rejected at the 0.05 level of significance, it must be rejected at the 0.025 level.

Marathon Runners A researcher wants to study the average miles run per day for marathon runners.In testing the hypotheses: H₀: $\mu$ = 25 miles vs.H₁: $\mu$ $\neq$ 25 miles, a random sample of 36 marathon runners drawn from a normal population whose standard deviation is 10, produced a mean of 22.8 miles weekly. -{Marathon Runners Narrative} Compute the value of the test statistic and specify the rejection region associated with 5% significance level.

A random sample of 100 observations from a normal population whose standard deviation is 50 produced a mean of 75.Does this statistic provide sufficient evidence at the 5% level of significance to infer that the population mean is not 80?

The owner of a local nightclub has recently surveyed a random sample of n = 300 customers of the club.She would now like to determine whether or not the mean age of her customers is over 35.If so, she plans to alter the entertainment to appeal to an older crowd.If not, no entertainment changes will be made.Suppose she found that the sample mean was 35.5 years and the population standard deviation was 5 years.What is the p-value associated with the test statistic?

Which of the following p-values will lead us to reject the null hypothesis if the level of significance equals 0.05?

The ____________________ of a test is the probability of observing a test statistic at least as extreme as the one from your sample, given that H₀ is true.

Rechargeable Batteries A researcher wants to study the average lifetime of a certain brand of rechargeable batteries (in hours).In testing the hypotheses, H₀: $\mu$ = 950 hours vs.H₁: $\mu$$\neq$ 950 hours, a random sample of 25 rechargeable batteries is drawn from a normal population whose standard deviation is 200 hours. -{Rechargeable Batteries Narrative} Review the results of the previous questions.What is the effect of decreasing the significance level on the value on $\beta$ ?

Production Filling A production filling operation has a historical standard deviation of 6 ounces.When in proper adjustment, the mean filling weight for the production process is 50 ounces.A quality control inspector periodically selects at random 36 containers and uses the sample mean filling weight to see if the process is in proper adjustment. -{Production Filling Narrative} State the null and alternative hypotheses.

A Type II error is represented by $\beta$ ; it is the probability of failing to reject a false null hypothesis.

For a given sample size n, if the level of significance $\alpha$ is decreased, the power of the test will:

The power of a test plus the probability of a Type II error equals ____________________.

Toaster Oven An appliance manufacturer claims to have developed a new toaster oven that consumes an average of no more than 250 W.From previous studies, it is believed that power consumption for toaster ovens is normally distributed with a standard deviation of 18 W.A consumer group suspects the actual average is more than 250 W.They take a sample of 20 toaster ovens and calculate the average consumption to be 260 W. -{Toaster Oven Narrative} Calculate the p-value of the test.

For a given level of significance, if the sample size is increased, the probability of committing a Type II error will decrease.

For a given level of significance $\alpha$ , if the sample size n is increased, the probability of a Type II error $\beta$ will:

LSAT Scores The Admissions officer for the graduate programs at the University of Pennsylvania believes that the average score on the LSAT exam at his university is significantly higher than the national average of 1,300.An accepted standard deviation for LSAT scores is 125.A random sample of 25 scores had an average of 1,375. -{LSAT Scores Narrative} Calculate the p-value.

If the p value is less than $\alpha$ in a two-tail test:

If the probability of committing a Type I error for a given test is decreased, then for a fixed sample size n, the probability of committing a Type II error will:

If a researcher fails to reject a true null hypothesis, he has made a(n) ____________________ decision.