Math Problem Statement

You are conducting a study to see if the proportion of men over the age of 50 who regularly have their prostate examined is significantly less than 0.37. A random sample of 797 men over the age of 50 found that 199 have their prostate regularly examined. Do the sample data provide convincing evidence to support the claim? Test the relevant hypotheses using a 7% level of significance.

Solution

To find the correct test statistic, you need to use the formula for the z-test for a proportion:

z=p^p0p0(1p0)nz = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}

Where:

  • p^=\hat{p} = sample proportion = 1997970.25\frac{199}{797} \approx 0.25
  • p0=p_0 = population proportion = 0.37
  • n=n = sample size = 797

Plug these values into the formula:

z=0.250.370.37(10.37)797z = \frac{0.25 - 0.37}{\sqrt{\frac{0.37(1-0.37)}{797}}}

Calculate the standard error:

SE=0.37×0.637970.01747\text{SE} = \sqrt{\frac{0.37 \times 0.63}{797}} \approx 0.01747

Calculate the z-score:

z=0.250.370.017476.88z = \frac{0.25 - 0.37}{0.01747} \approx -6.88

So the correct test statistic, rounded to two decimals, is approximately 6.88-6.88.

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Math Problem Analysis

Mathematical Concepts

Hypothesis Testing
Proportions
Standard Normal Distribution

Formulas

z = (p̂ - p₀) / √(p₀(1-p₀)/n)

Theorems

Central Limit Theorem

Suitable Grade Level

College/Undergraduate Statistics