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Exam 13: Hypothesis Testing: Describing a Single Population

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Question 61

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Determine the p-value associated with each of the following values of the standardised test statistic z. a. Two-tail test, z = 1.50. b. One-tail test, z = 1.05. c. One-tail test, z = -2.40.

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a. 0.1336....

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Question 62

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In testing the hypotheses $H _ { 0 } : \mu = 50$ . $H _ { 1 } : \mu < 50$ . we found that the standardised test statistic is z = -1.59. Calculate the p-value.

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Question 63

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In testing the hypotheses: $H _ { 0 } : \mu = 40$ $H _ { 1 } : \mu \neq 40,$ the following information was given: $\sigma = 5.5 , \quad n = 25 , \quad \bar { x } = 42 , \quad \alpha = 0.10$ , and the sampled population is normally distributed. a. Calculate the value of the test statistic. b. Set up the rejection region. c. Determine the p-value. d. Interpret the result.

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.

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.

In order to determine the p-value, which of the following items of information is not needed? A. The level of significance. B. Whether the test is one- or two-tailed. C. The value of the test statistic. D. All of the above are needed.

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 testing the hypotheses: $H _ { 0 } : \mu =$ 35 H₁ : $\mu$ < 35, The following information is known: n = 49, $\bar { x }$ = 37 and $\sigma$ = 16. The standardised test statistic equals: $\begin{array}{|l|l|}\hline \text { A. } & 0.33 . \\\hline \text { B. } & -0.33 \\\hline \text { C. } & -2.33 \\\hline \text { D. } & 2.33 . \\\hline\end{array}$

The p-value of a test is the smallest value of $\alpha$ at which the null hypothesis can be rejected.

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. Calculate the p-value.

The admissions officer for the graduate programs at the University of Adelaide believes that the average score on an exam at his university is significantly higher than the national average of 1300. Assume that the population standard deviation is 125 and that a random sample of 25 scores had an average of 1375. a. State the appropriate null and alternative hypotheses. b. Calculate the value of the test statistic and set up the rejection region. What is your conclusion? c. Calculate the p-value. d. Does the p-value confirm the conclusion in part (b)?

The probability of a Type II error is denoted by: $\begin{array}{|l|l|}\hline \text { A. } & \alpha . \\\hline \text { B. } & \beta \\\hline \text { C. } & 1-\alpha \\\hline \text { D. } & 1-\beta \\\hline\end{array}$

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.

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 $\beta$ = 0.2061. A) Calculate the power of the test. B) Interpret your answer to part (a).

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 and the power of the test is 0.3465 Interpret the meaning of the power in the previous question.

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 n is increased from 25 to 4, β = 0.5233. What is the effect of increasing the sample size on the value of $\beta$ ?

A spouse stated that the average amount of money spent on Christmas gifts for immediate family members is above $1200. The correct set of hypotheses is: $\begin{array}{|l|l|}\hline\text { A. } & H _ { 0 } : \mu = 200\\& H _ { 1 } : \mu < 1200 \\\hline \text { B. } & H _ { 0 } : \mu > 1200 \\& H _ { 1 } : \mu = 1200 \\\hline \text { C. } & H _ { 0 } : \mu = 1200 \\& H _ { 1 } : \mu > 1200 \\\hline \text { D. } & H _ { 0 } : \mu < 1200 \\& H _ { 1 } : \mu = 1200 \\\hline\end{array}$

Whenever the null hypothesis is not rejected: $\begin{array}{|l|l|}\hline A.&\text {the null hypothesis is true. }\\\hline B.&\text { the alternative hypothesis is false.}\\\hline C.&\text { the null hypothesis is maintained.}\\\hline D.&\text {the null hypothesis is accepted. }\\\hline \end{array}$

A one-tail p-value is two times the size of that for a two-tail test.