Example 1 - Georgia State University

Example 1

The null and alternative hypotheses are given. Determine whether the hypothesis test is left-tails, righttailed, or two-tailed. What parameter is being tested? H0: = 110 H1: > 110

This is an equal hypothesis versus a `greater than', so this is right-tailed.

Recall that is the population mean, is the population standard deviation, and p is a population proportion.

Since the hypotheses refer to we are testing the population standard deviation.

Example 2

Determine the null and alternative hypotheses, explain what it would mean to make a Type I error, and explain what it would mean to make a Type II error.

Three years ago, the mean price of a single-family home was $243,782. A real estate broker believes that the mean price has increased since then.

Either it has has stayed the same, or we want to test what the broker believes, which is that it has increased.

We are talking about the mean, .

So H0: = $243,782

H1: > $243,782

A Type I error is committed when the null hypothesis is rejected, when it is actually true.

So a Type I error would be that the broker rejects the hypothesis that the mean price is $243,782, when it is the true mean cost.

A Type II error is committed when the null hypothesis is not rejected when the alternative hypothesis is true.

A type II error would be if the broker fails to reject the hypothesis that the mean price is $243,782, when the true mean price is greater than $243,782.

Example 3

Suppose the null hypothesis is rejected. State the conclusion based on the results of the test.

Three years ago, the mean price of a single-family home was $243,782. A real estate broker believes that the mean price has increased since then.

The conclusion: There is sufficient evidence to conclude that the mean price of a single-family home has increased. (This is because the null hypothesis, which stated that it stayed the same, was rejected.)

Example 4

Suppose the null hypothesis is rejected. State the conclusion based on the results of the test.

Six years ago, 12.4% of registered births were to teenage mothers. A sociologist believes that the percentage has increased since then.

There is sufficient evidence to conclude that the percentage of teenage mothers has increased. (Because the null hypothesis was rejected, so the alternative hypothesis that it has increased, is chosen.)

Example 5

Five years ago, 10.2% of high school students had tried marijuana for the first time before age 13. A school resource officer (SRO) thinks that the proportion of high school students who have tried marijuana for the first time before the age of 13 has decreased since then.

Determine the null and alternative hypotheses.

This is a proportion (not a mean or standard deviation). 10.2% as a decimal is 0.102. We either test that it's the same or that it has decreased (is less than).

H0: p = 0.102

H1: p < 0.102

Suppose sample data indicate that the null hypothesis should be rejected. State the conclusion of the researcher.

There is sufficient evidence to conclude that the proportion of high school students has decreased. (Because we rejected the null hypothesis that it did not change.)

Suppose, in fact, that the proportion of high school students who have tried marijuana before the age of 13 was 10.2%. Was a type I or type II error committed?

This is a Type I error because he rejected the null hypothesis, when in fact, it was true.

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