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Chapter 9: Q.79 (page 589)

Pressing pills A drug manufacturer forms tablets by compressing a granular material that contains the active ingredient and various fillers. The hardness of a sample from each batch of tablets produced is measured to control the compression process. The target value for the hardness is μ=11.5. The hardness data for a random sample of 20 tablets are

11.627 11.613 11.493 11.602 11.360

11.374 11.592 11.458 11.552 11.463

11.383 11.715 11.485 11.509 11.429

11.477 11.570 11.623 11.472 11.531

Is there significant evidence at the 5% level that the mean hardness of the tablets differs from the target value? Carry out an appropriate test to support your answer.

Short Answer

Expert verified

the mean hardness of the tablets does not differ from the target value.

Step by step solution

01

Given Information

The hardness data for a random sample of 20tablets are

11.627    11.613    11.493    11.602    11.36011.374    11.592    11.458    11.552    11.46311.383    11.715    11.485    11.509    11.42911.477    11.570    11.623    11.472    11.531
02

Explanation

From the data given,

x-=11.516

μ=11.5

we calculate the null and alternative hypotheses,

H0:μ=11.5Ha:μ≠11.5

Using,

localid="1652942386796" t=x-−μsn=11.516−11.500.095020=0.77

The test statistic =0.77

Hence the mean hardness of the tablets does not differ from the target value μ=11.5as the critical value2.093 is greater than the test statistic 0.77.

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Most popular questions from this chapter

Improving health A large company's medical director launches a health promotion campaign to encourage employees to exercise more and eat better foods. One measure of the effectiveness of such a program is a drop in blood pressure. The director chooses a random sample of 50employees and compares their blood pressures from physical cams given before the campaign and again a year later. The mean change (after - before) in systolic blood pressure for these 50employees is -6and the standard deviation is 19.8.

(a) Do these data provide convincing evidence of an average decrease in blood pressure among all of the company's employees during this year? Carry out a test at the α=0.05significance level.

(b) Can we conclude that the health campaign caused a decrease in blood pressure? Why or why not?

In planning a study of the birth weights of babies whose mothers did not see a

doctor before delivery, a researcher states the hypotheses as

H0:μ<1000gramsHa:μ=900grams

Slow response times by paramedics, firefighters, and policemen can have serious consequences for accident victims. In the case of life-threatening injuries, victims generally need medical attention within 8 minutes of the accident. Several cities have begun to monitor emergency response times. In one such city, the mean response time to all accidents involving life-threatening injuries last year was M 6.7 minutes. Emergency personnel arrived within 8 minutes after 78% of all calls involving life-threatening injuries last year. The city manager shares this information and encourages these first responders to “do better.” At the end of the year, the city manager selects an SRS of 400 calls involving life-threatening injuries and examines the response times.

(a) State hypotheses for a significance test to determine whether the average response time has decreased. Be sure to define the parameter of interest.

(b) Describe a Type I error and a Type II error in this setting, and explain the consequences of each.

(c) Which is more serious in this setting: a Type I error or a Type II error? Justify your answer.

The z statistic for a test of H0 :p = 0.4 versus Ha : p > 0.4 is z = 2.43. This test is

(a) not significant at either α=0.05or α=0.01.

(b) significant at α=0.05but not at α=0.01

(c) significant at α=0.01but not at α=0.05.

(d) significant at both α=0.05and α=0.01.

(e) inconclusive because we don’t know the value of ˆp .

Tests and CIs The P-value for a one-sided test of the null hypothesisH0:μ=15is 0.03.

(a) Does the 99%confidence interval for μinclude 15? Why or why not?

(b) Does the 95%confidence interval for μinclude 15? Why or why not?
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