g Suppose the population 10-year cumulative incidence of prostate cancer among 70-year old men is 0.06. Further, suppose 260 men without prostate cancer, aged 50 years were randomly sampled and 13 of them developed prostate cancer within 10 years. Is there evidence at the 0.05 significance level to conclude that the cumulative incidence of prostate cancer among 50-year old men differs from that of 70-year old men

Answer:

There is not enough statistical evidence to conclude that the cumulative incidence of prostrate cancer among 50-year old men differs from that of 70-year old men

Step-by-step explanation:

The population 10 year cumulative incidence of prostrate cancer among 70 year old men, p = 0.06

The number of 50 year old men in the sample, n = 260 men

The number of the sampled men that developed prostrate cancer = 13 men

The significance level, α = 0.05

Let the null hypothesis, H₀: [tex]\hat{p}[/tex] = p

The alternative hypothesis, Hₐ: [tex]\hat{p}[/tex] ≠ p

The standard score, [tex]z_{\alpha /2}[/tex] = 1.96

The test statistic, 'z', is given as follows;

[tex]z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p \cdot q}{n}}}[/tex]

[tex]\hat{p}[/tex] = 13/260 = 0.05

q = 1 - p = 1 - 0.06 = 0.94

[tex]Therefore, \ z=\dfrac{0.05-0.06}{\sqrt{\dfrac{0.06 \times 0.94}{260}}} = \dfrac{\sqrt{9165} }{141} \approx -0.68[/tex]

From the z-table, we find the p-value as follows;

P(z ≈ -0.68) = 0.24825

Therefore, given that the p-value, 0.24825 is larger than the significance level, α/2 = 0.025, we fail to reject the null hypothesis, and therefore, there is not enough statistical evidence to conclude that the cumulative incidence of prostrate cancer among 50-year old men differs from that of 70-year old men

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