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Do the data provide convincing evidence of a difference in the effectiveness of the four treatments? Carry out an appropriate test at the =0.05significance level.

Short Answer

Expert verified

Yes, there is sufficient evidence that there is an association between the variables.

Step by step solution

01

Given information

The given data is

02

Explanation

Observed counts

The expected counts are the row total multiplied by the column total, divided by the sample sizen=6602

The chi-square statistic is the sum of squared deviations (between observed and expected counts) divided by the expected count:

localid="1650644880819" 2=(250-205.8128)2205.8128+(206-205.8128)2205.8128+(211-206.4368)2206.4368+(157-205.9376)2205.9376+(1399-1443.1872)21443.1872+(1443-1443.1872)21443.1872+(1443-1447.5632)21447.5632+(1493-1444.0624)21444.0624=24.243

The interval for the P-value can then be found in table C in the column title which have the X2-value in the corresponding interval in the row with

localid="1650644859112" df=(c-1)(r-1)=(4-1)(2-1)=3

The two tailed pvalue is 0.000044

P<0.0005

If the P-value is smaller than the significance level, then the null hypothesis is rejected.

P<0.05RejectH0

There is sufficient evidence that there is an association between the variables.

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Most popular questions from this chapter

The chi-square statistic is

(a) (18-25)225+(22-25)225+(39-25)225+(21-25)225

(b) (25-18)218+(25-22)222+(25-39)239+(25-21)221

(c) (18-25)25+(22-25)25+(39-25)25+(21-25)25

(d)(18-25)2100+(22-25)2100+(39-25)2100+(21-25)2100

(e)(0.18-0.25)20.25+(0.22-0.25)20.25+(0.39-0.25)20.25+(0.21-0.25)20.25

Reading and grades (10.2) Summary statistics for the two groups from Minitab are provided below.

(a) Explain why it is acceptable to use two-sample t procedures in this setting.

(b) Construct and interpret a 95% confidence interval for the difference in the mean English grade for light and heavy readers.

(c) Does the interval in part (b) provide convincing evidence that reading more causes an increase in students鈥 English grades? Justify your answer.

Benford鈥檚 lawFaked numbers in tax returns, invoices, or expense account claims often display patterns that aren鈥檛 present in legitimate records. Some patterns are obvious and easily avoided by a clever crook. Others are more subtle. It is a striking fact that the 铿乺st digits of numbers in legitimate records often follow a model known as Benford鈥檚 law.3 Call the 铿乺st digit of a randomly chosen record X for short. Benford鈥檚 law gives this probability model for X (note that a 铿乺st digit can鈥檛 be 0):

A forensic accountant who is familiar with Benford鈥檚 law inspects a random sample of invoices from a company that is accused of committing fraud. The table below displays the sample data.

(a) Are these data inconsistent with Benford鈥檚 law? Carry out an appropriate test at the =0.05level to support your answer. If you 铿乶d a signi铿乧ant result, perform follow-up analysis.

(b) Describe a Type I error and a Type II error in this setting, and give a possible consequence of each. Which do you think is more serious?

The conditions for carrying out the chi-square test in exercise T11.2 are

I. Separate random samples from the populations of interest.

II. Expected counts large enough.

III. The samples themselves and the individual observations in each sample are independent.

Which of the conditions is (are) satisfied in this case?

(a) I only

(d) Il and III only

(b) II only

(e) I, II, and III

(c) I and II only

How to quit smoking It鈥檚 hard for smokers to quit. Perhaps prescribing a drug to fight depression will work as well as the usual nicotine patch. Perhaps combining the patch and the drug will work better than either treatment alone. Here are data from a randomized, double-blind trial that compared four treatments. A 鈥渟uccess鈥 means that the subject did not smoke for a year following the beginning of the study.

Group Treatment Subjects Successes

1 Nicotine patch 244 40

2 Drug 244 74

3 Patch plus drug 245 87

4 Placebo 160 25

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(c) Explain in words what the null hypothesis H0: p1 = p2 = p3 = p4 says about subjects鈥 smoking habits.

(d) Find the expected counts if H0 is true, and display them in a two-way table similar to the table of observed counts.

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