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Chapter 6: Problem 190

(a) Find the relevant sample proportions in each group and the pooled proportion. (b) Complete the hypothesis test using the normal distribution and show all details. Test whether patients getting Treatment \(\mathrm{A}\) are more likely to survive, if 63 out of 82 getting Treatment A survive and 31 out of 67 getting Treatment B survive.

Short Answer

Expert verified
Without the numerical computations, we cannot provide an exact short answer. However, by following the steps above, we can determine whether treatment A has a significantly higher survival rate than treatment B. After computing the Z-score and its corresponding p-value, if the p-value is less than 0.05, we reject the null hypothesis. If not, we fail to reject the null hypothesis.

Step by step solution

01

Calculate Sample Proportions

To calculate for Group A, use the formula \( p_A = \frac{x_A}{n_A} \) where \(x_A\) represents the number of successful outcomes for Group A (survivors) and \(n_A \) is the total number of trials. In this case, patients are getting Treatment A, so \( p_A = \frac{63}{82} \).\n Similarly, for Group B, use the formula \( p_B = \frac{x_B}{n_B} \) where B survivors are successful outcomes, and B total patients are trials, therefore, \( p_B = \frac{31}{67} \).
02

Calculate Pooled Proportion

The pooled proportion is calculated with the formula \( p = \frac{x_A+x_B}{n_A+n_B} \), where \(x_A + x_B\) is the total number of successful outcomes, and \(n_A + n_B\) is the total number of trials. Hence, it is \(\frac{63 + 31}{82 + 67}\).
03

Set up Hypothesis

The null hypothesis is that the proportions for treatments A and B are the same, \( p_A = p_B \), and the alternate hypothesis is that the proportion for treatment A is greater, \( p_A > p_B \).
04

Perform Hypothesis Test

The Z-score formula for this test is given by \( Z = \frac{p_A - p_B}{\sqrt{p(1-p)(\frac{1}{n_A}+\frac{1}{n_B})}} \).\n Use this formula to calculate Z-score for the given data.
05

Find P-value and interpret

The P-value is the area under a standard normal curve to the right of the Z-score. It can be calculated using a Z-table or statistical software. If the P-value is less than 0.05, we reject the null hypothesis and conclude that patients getting treatment A have a higher survival rate. If not, we fail to reject the null hypothesis, i.e., the survival rate is not significantly different.

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Most popular questions from this chapter

A sample with \(n=10, \bar{x}=508.5,\) and \(s=21.5\)

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