Exam 11: Introduction to Hypothesis Testing

The power of the test refers to the probability of rejecting a false null hypothesis.

You reject H₀ if the p-value of your hypothesis is ____________________ than the significance level.

If the value of the sample mean is close enough to the hypothesized value μ₀ of the population mean μ,then:

The probability of a Type II error is denoted by ____________________.

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 value of the test statistic and set up the rejection region at the 0.025 level.What is your conclusion?

If a hypothesis is not rejected at the 0.10 level of significance,it:

Increasing the probability of a Type I error will increase the probability of a Type II error.

During the last energy crisis,a government official claimed that the average car owner refills the tank when there is more than 3 gallons left.To check the claim,10 cars were surveyed as they entered a gas station.The amount of gas remaining before refill was measured and recorded as follows (in gallons): 3,5,3,2,3,3,2,6,4,and 1.Assume that the amount of gas remaining in tanks is normally distributed with a standard deviation of 1 gallon.Compute the probability of a Type II error and the power of the test if the true average amount of gas remaining in tanks is 3.5 gallons anda= 0.10.

In order to determine the p-value,it is necessary to know the level of significance.

The p-value is the probability that the null hypothesis is true.

In a criminal trial,a Type I error is made when an innocent person is convicted.

There is an inverse relationship between the probabilities of Type I and Type II errors; as one increases,the other decreases,and vice versa.

Marathon Runners A researcher wants to study the average miles run per day for marathon runners.In testing the hypotheses: H₀: μ = 25 miles vs.H₁: μ ≠ 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 p-value.

If your p-value is greater than 0.900 you should reject H₀ at the 0.10 level.

We have created a 90% confidence interval for μ with the result (25,32).What conclusion will we make if we test H₀: μ = 28 vs.H₁: μ ≠ 28 at α = 0.10?

Explain why a Type I error and a Type II error have an inverse relationship.

The owner of a local Jazz Club has recently surveyed a random sample of n = 200 customers of the club.She would now like to determine whether or not the mean age of her customers is over 30.If so,she plans to alter the entertainment to appeal to an older crowd.If not,no entertainment changes will be made.The appropriate hypotheses to test are:

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} Use the p-value to test the hypotheses.

Suppose that we reject a null hypothesis at the 0.05 level of significance.Then for which of the following α-values do we also reject the null hypothesis?

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