Question

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 780 men over the age of 50 found that 206 have their prostate regularly examined. Do the sample data provide convincing evidence to support the claim. Test the relevant hypotheses using a 10% level of significance. Give answers to at least 4 decimal places. What are the correct hypotheses?

Answer 1

Answer:

1. Yes, the sample data provides convincing evidence to support the claim that the proportion of men over the age of 50 who regularly have their prostate examined is significantly less than 0.37

2. The correct hypotheses are;

The null hypothesis is H₀: p ≥ p₀

The alternative hypothesis is Hₐ: p < p₀

Step-by-step explanation:

given

The null hypothesis is H₀: p ≥ p₀ where p₀ = 0.37

The alternative hypothesis is Hₐ: p < p₀

The formula for the z test is presented as follows;

$z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0 (1 - p_0)}{n}}}$

Where:

$\hat p$ = Sample proportion = 206/780 = 0.264

p₀ = Population proportion = 0.37

n = Sample size = 780

α = Significance level = 10% = 0.01

Plugging in the values, we have;

$z=\dfrac{0.264-0.37}{\sqrt{\dfrac{0.37 (1 - 0.37)}{780}}} = -6.13$

From the the z relation/computation, we have the p value = 0.000000000451

Since the p value which is 0.000000000451 is less than α, which is 0.01 we reject the null hypothesis and we fail to reject the alternative hypothesis, that is there is sufficient statistical evidence to suggest that the proportion of men over the age of 50 who regularly have their prostate examined is significantly less than 0.37.