Variance and standard error

Question

A hospital treated 412 skin cancer patients over a year. Of these, 197 were female.

Give the point estimate of the proportion of females seeking treatment for skin cancer.

Give estimates of the variance and standard error of the point estimate.

Give a 95% confidence interval for the population proportion of females seeing treatment for skin cancer.

Use an appropriate test to determine whether this dataset provides statistically significant evidence that males are more likely to seek treatment for skin cancer.

Answer 1

Given:

n: Total number of patients = 412
x₁: Number of females in the sample seeking cancer treatment = 197
x₂: Number of males in the sample seeking cancer treatment = 412-197 = 215
   
Define:

p₁: sample proportion of females seeking cancer treatment.

p₁= x₁/n = 197/412 = 0.4782

p₁: sample proportion of males seeking cancer treatment.

p₂ = x₂/n = 215/412 = 0.5218

We know that the sample proportion is an unbiased estimator of population proportion, hence the proportion of females seeking treatment for skin cancer is

p ^= p₁=0.4782

In case of proportions, the population variance is estimated by:

The standard error is nothing but the square root of variance
Hence,

The 95% confidence interval for the population proportion of females seeing treatment for skin cancer is given by:

Where, Z_C (Critical value) = 1.96
Hence,

This is the required confidence interval.

Now we are supposed to test whether males are more likely to seek treatment for skin cancer.

Null hypothesis:

H₀: There is no significant difference in number of cancer patients due to according to gender
H₀: P₁ = P₂

Alternative hypothesis:

H_a: Males are more likely to seek treatment for skin cancer.
H_a: P₁ < P₂

α (level of significance) = 0.05         One tailed test
Z_α (Critical value) = -1.64

Assumptions:

The two samples come from independent population.
Population is normally distributed.

Test Statistic:

Where,
 
P = 1/2

Q= 1/2

Hence Z = - 1.2541

 P value = P (Z < Z observed)
             = P (Z < -1.2541 )
             = 0.1049

Decision Rule:

Reject H₀ if P value is less than the level of significance.

Decision:

Since observed value (-1.2541) > critical value (-1.64) and P value (observed level of significance) = 0.1049 is greater than α (level of significance) = .05, we fail to reject H₀.

Conclusion:

There is no significant difference in number of cancer patients due to according to gender.