Exam 13: Hypothesis Testing: Describing a Single Population

A test for the population mean $\mu$ produces a test-statistic z = -0.75. The p-value associated with the test is 0.2266 if the test is a left-tail test, it is 0.7734 if the test is a right-tail test, and it is 0.4533 if the test is a two-tail test.

A null hypothesis is a statement about the value of a population parameter; it is put up for testing in the face of numerical evidence.

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 Recalculate $\beta$ if n is increased from 25 to 40.

The probability of a Type I error is denoted by:

Which of the following test statistics may be used to test a value of the population proportion?

In any given test, it is possible to commit the Type I and Type II errors at the same time.

A Type I error is represented by $\alpha$ , and is the probability of incorrectly rejecting a true null hypothesis.

A random sample of 100 families in a large city revealed that on the average these families have been living in their current homes for 35 months. From previous analyses, we know that the population standard deviation is 30 months and that ? = 0.2061. a. Recalculate $\beta$ if $\alpha$ is lowered from 0.05 to 0.01. b. What is the effect of decreasing the significance level on the value of $\beta$ ?

When testing a value of the population proportion, we must do a Z-test.

If we do not reject the null hypothesis, we conclude that there is enough statistical evidence to infer that the null hypothesis is true.

The probability of making a Type I error and the level of significance are the same.

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

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 Recalculate $\beta$ if $\alpha$ is lowered from 0.10 to 0.05.

When testing a value of the population mean, if the population variance is unknown, then we must do a t-test

In testing the hypotheses: $H _ { 0 } : \mu = 25$ $H _ { 1 } : \mu \neq 25$ , a random sample of 36 observations drawn from a normal population whose standard deviation is 10 produced a mean of 22.8. Explain briefly how to use the confidence interval in the previous question to test the hypothesis.

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 $\mu$ :

A social scientist claims that the average adult watches less than 26 hours of television per week. He collects data on 25 individuals' television viewing habits, and finds that the mean number of hours that the 25 people spent watching television was 22.4 hours. If the population standard deviation is known to be 8 hours, can we conclude at the 1% significance level that he is right?

Whenever the null hypothesis is not rejected:

Calculate the probability of a Type II error for the following test of hypothesis: $H _ { 0 } : \mu = 50$ $H _ { 1 } : \mu > 50$ given that $\mu = 55$ , $\alpha = 0.05 , \sigma = 10$ and n = 16.

In testing the hypotheses: $H _ { 0 } : \mu = 25$ $H _ { 1 } : \mu \neq 25$ , a random sample of 36 observations was drawn from a normal population. The sample standard deviation is 10 and the sample mean is 22.8. Can we conclude at the 5% significance level that the population mean is not significantly different to 25?