Exam 13: Hypothesis Testing: Describing a Single Population

Using the confidence interval when conducting a two-tail test for the population mean $\mu$ , we do not reject the null hypothesis if the hypothesised value for $\mu$ is smaller than the upper confidence limit.

Reducing the probability of a Type I error, increases the probability of a Type II error.

The p-value criterion for hypothesis testing is to retain the null hypothesis if:

In any test, the probability of a Type I error and the probability of a Type II error add up to 1.

In a criminal trial, a Type II error is made when:

The manager of a sports store is considering trading on Public holidays. She believes that a majority of consumers would consider visiting the sports store on a Public holiday. She takes a random sample of 50 customers and finds that 29 would visit the sports store on a Public holiday. Is there significant evidence to support the manager's claim? Test at the 5% level of significance.

A professor of statistics refutes the claim that the average student spends 6 hours studying for the final. To test the claim, the hypotheses H₀: $\mu$ = 6, H₁: $\mu$ < 6 should be used.

If the p-value for a one tailed test is 0.03, would you have the same conclusion at a significance level of 0.05 and at a significance level of 0.10?

The p-value is usually 0.05.

Formulate the null and alternative hypotheses for each of the following statements: a. The average Australian household owns 2.5 cars. b. A researcher at the University of Adelaide is looking for evidence to conclude that the majority of students drive to university. c. The manager of the University of Tasmania bookstore claims that the average student spends less than $400 per semester at the university's bookstore.

The manager of a fast food restaurant is investigating the average number of fries per Large-sized offered to customers, to assess quality control. The manager took a random sample of five Large-sized orders of fries, and counted the number of fries per serve, with the results given below: 73 75 83 68 78 Assume that the number of French fries served at this fast food restaurant is normally distributed. Can we infer at the 5% significance level that the average number of fries served in a Large-sized order of fries at this fast food restaurant is over 70?

During the Gulf War, a government official claimed that the average car owner refilled the fuel tank when there was more than 3 litres of petrol left. To check the claim, 10 cars were surveyed as they entered a service station. The amount of petrol (in litres) was measured and recorded as shown below. 3 5 3 2 3 3 2 6 4 1 Assume that the amount of petrol remaining in the tanks is normally distributed with a standard deviation of 1 litre. Can we conclude at the 10% significance level that the official was correct?

Consider the hypotheses $H_ { 0 } : \mu = 950$ $H _ { 1 } : \mu \neq 950$ . Assume that $\mu = 1000$ $\alpha = 0.10$ $\sigma = 200$ , and n = 25 and $\beta$ = 0.6535 Calculate the power of the test.

A Type II error is committed if we make:

When testing whether the majority of voters in an electorate will vote for a particular candidate, which of the following sets of hypotheses are correct?

If we reject a null hypothesis at the 0.05 level of significance, then we must also reject it at the 0.10 level.

Consider the hypotheses $H _ { 0 } : \mu = 950$ $H _ { 1 } : \mu \neq 950$ . Assume that $\mu = 1000$ $\alpha = 0.10$ $\sigma = 200$ , and n = 25 and $\beta$ = 0.6535 When we recalculate $\beta$ if $\alpha$ is lowered from 0.10 to 0.05, β = 0.7604 What is the effect of decreasing the significance level on the value of $\beta$ ?

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

There is a direct relationship between the power of a test and the probability of a Type II error.

In a one-tail test for the population mean, if the null hypothesis is not rejected when the alternative hypothesis is true, a Type I error is committed.