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Chapter 8: Q3E (page 325)

For which of the given P-values would the null hypothesis be rejected when performing a level .05 test?

a. .001 b. .021 c. .078

d..047 e. .148

Short Answer

Expert verified

a) Reject the null hypothesis\({H_0}\).

b) Reject the null hypothesis\({H_0}\).

c) Fail to reject the null hypothesis\({H_0}\).

d) Reject the null hypothesis\({H_0}\).

e) Fail to reject the null hypothesis \({H_0}\).

Step by step solution

01

Step 1:Null hypothesis.

The null hypothesis, denoted by H0, is the claim that is initially assumed to be true (the 鈥減rior belief鈥� claim). The alternative hypothesis, denoted by Ha, is the assertion that is contradictory to H0.

The null hypothesis will be rejected in favor of the alternative hypothesis only if sample evidence suggests that H0 is false. If the sample does not strongly contradict H0, we will continue to believe in the plausibility of the null hypothesis. The two possible conclusions from a hypothesis-testing analysis are then reject H0 or fail to reject H0.

02

Step 2:Solution for part a).

Given that,

\(\begin{array}{l}P = 0.001\\\alpha = 0.05\end{array}\)

If the \(P\)-value is less than the significance level \(\alpha \), then reject the null hypothesis

\({H_0}\), else we fail to reject the null hypothesis \({H_0}\).

\(P = 0.001 < 0.05\)

In the given \(P < \alpha \), so, it rejects the \({H_0}\).

Reject the null hypothesis \({H_0}\).

03

Step 3:Solution for part b).

Given that,

\(\begin{array}{l}P = 0.021\\\alpha = 0.05\end{array}\)

If the \(P\)-value is less than the significance level \(\alpha \), then reject the null hypothesis \({H_0}\), else we fail to reject the null hypothesis \({H_0}\).

\(P = 0.021 < 0.05\)

In the given \(P < \alpha \), so, it rejects the \({H_0}\).

Reject the null hypothesis \({H_0}\).

04

Step 4:Solution for part c).

Given that,

\(\begin{array}{l}P = 0.078\\\alpha = 0.05\end{array}\)

If the \(P\)-value is less than the significance level \(\alpha \), then reject the null hypothesis \({H_0}\), else we fail to reject the null hypothesis \({H_0}\).

\(P = 0.078 > 0.05\)

In the given \(P > \alpha \), so, it fails to rejects the \({H_0}\).

Fail to reject the null hypothesis \({H_0}\).

05

Step 5:Solution for part d).

Given that,

\(\begin{array}{l}P = 0.047\\\alpha = 0.05\end{array}\)

If the \(P\)-value is less than the significance level \(\alpha \), then reject the null hypothesis \({H_0}\), else we fail to reject the null hypothesis \({H_0}\).

\(P = 0.047 < 0.05\)

In the given \(P < \alpha \), so, it rejects the \({H_0}\).

Reject the null hypothesis \({H_0}\).

06

Step 6:Solution for part e).

Given that,

\(\begin{array}{l}P = 0.148\\\alpha = 0.05\end{array}\)

If the \(P\)-value is less than the significance level \(\alpha \), then reject the null hypothesis \({H_0}\), else we fail to reject the null hypothesis \({H_0}\).

\(P = 0.148 > 0.05\)

In the given \(P > \alpha \), so, it fails to rejects the \({H_0}\).

Fail to reject the null hypothesis \({H_0}\).

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Most popular questions from this chapter

Reconsider the paint-drying situation of Example 8.5, in which drying time for a test specimen is normally distributed with 蟽 = 9. The hypotheses H0: 碌 =75 versus Ha: 碌 <75 are to be tested using a random sample of n= 25 observations.

a.How many standard deviations (of X) below the null value is \(\overline x = 72.3\)?

b.If \(\overline x = 72.3\), what is the conclusion using 伪 =.002?

c.For the test procedure with 伪 =.002, what is 尾(70)?

d.If the test procedure with 伪 =.002 is used, what n is necessary to ensure that 尾(70) = .01?

e.If a level .01 test is used with n5 100, what is the probability of a type I error when m5 76?Answer the following questions for the tire problem in Example 8.7.

a.If \(\overline x = 30,960\) 30,960 and a level 伪=.01 test is used, what is the decision?

b.If a level .01 test is used, what is 尾(30,500)?

c.If a level .01 test is used and it is also required that 尾(30,500) = .05, what sample size n is necessary?

d.If \(\overline x = 30,960\), what is the smallest 伪 at which H0 can be rejected (based on n = 16)?

With domestic sources of building supplies running low several years ago, roughly 60,000 homes were built with imported Chinese drywall. According to the article 鈥淩eport Links Chinese Drywall to Home Problems鈥� (New York Times, Nov. 24, 2009), federal investigators identified a strong association between chemicals in the drywall and electrical problems, and there is also strong evidence of respiratory difficulties due to the emission of hydrogen sulphide gas. An extensive examination of \(51\) homes found that \(41\) had such problems. Suppose these \(51\) were randomly sampled from the population of all homes having Chinese drywall.

a. Does the data provide strong evidence for concluding that more than \(50\% \) of all homes with Chinese drywall have electrical/environmental problems? Carry out a test of hypotheses using \(\alpha = .10\).

b. Calculate a lower confidence bound using a confidence level of \(99\% \) for the percentage of all such homes that have electrical/environmental problems.

c. If it is actually the case that \(80\% \) of all such homes have problems, how likely is it that the test of (a) would not conclude that more than \(50\% \) do?

A sample of n sludge specimens is selected and the pH of each one is determined. The one-sample t test will then be used to see if there is compelling evidence for concluding that true average pH is less than 7.0. What conclusion is appropriate in each of the following situations?

a.n= 6, t= -2.3, 伪= .05

b.n= 15, t= -3.1伪=.01

c.n= 12, t= -1.3, 伪= .05

d.n= 6, t = .7, 伪 = .05

e.n= 6, \(\overline x = 6.68,s/\sqrt n = .0820\)

Let 碌 denote the true average reaction time to a certain stimulus. For a z test of H0: 碌 =5 versus Ha: 碌 > 5, determine the P-value for each of the following values of the z test statistic.

a.1.42 b. .90 c. 1.96 d. 2.48 e. -.11

To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of welds is selected, and tests are conducted on each weld in the sample. Weld strength is measured as the force required to break the weld. Suppose the specifications state that mean strength of welds should exceed 100 lb/in2 ; the inspection team decides to test H0: 碌= 100 versus Ha: 碌> 100. Explain why it might be preferable to use

this Ha rather than 碌 < 100.

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