Complete Question
In a random sample of 700 men tested for the coronavirus, 63 were positive. Another independent random sample of 2950 women tested for the coronavirus resulted in 7 positive cases.Construct the 95% confidence interval for the difference between the positive rates of men and women
Answer:
The 95% confidence interval is [tex]0.833< p_1 - p_2 < 0.1059[/tex]
Step-by-step explanation:
From the question we are told that
The sample size of men is [tex]n_1 = 700[/tex]
The number men that tested positive is [tex]x_1 = 63[/tex]
The sample size of women is [tex]n_2 = 2950[/tex]
The number of women that tested positive is [tex]x_2 = 7[/tex]
From the question we are told the confidence level is 95% , hence the level of significance is
[tex]\alpha = (100 - 95 ) \%[/tex]
=> [tex]\alpha = 0.05[/tex]
Generally from the normal distribution table the critical value of [tex]\frac{\alpha }{2}[/tex] is
[tex]Z_{\frac{\alpha }{2} } = 1.96[/tex]
Generally the proportion of men that tested positive is mathematically represented as
[tex]\^ p_1 = \frac{ x_1 }{n_1}[/tex]
=> [tex]\^ p_1 = \frac{ 68 }{700}[/tex]
=> [tex]\^ p_1 = 0.097[/tex]
Generally the proportion of women that tested positive is mathematically represented as
[tex]\^ p_2 = \frac{ x_2 }{n_2}[/tex]
=> [tex]\^ p_2 = \frac{ 7 }{2950}[/tex]
=> [tex]\^ p_2 = 0.00237[/tex]
Generally the pooled population proportion is mathematically represented as
[tex]\^ p = \frac{x_1 + x_2 }{ n_1 + n_2}[/tex]
=> [tex]\^ p = \frac{ 63 + 7 }{ 700 + 2950}[/tex]
=> [tex]\^ p = 0.0192[/tex]
Generally the standard error is mathematically represented as
[tex]SE = \sqrt{\^ p (1- \^ p ) [ \frac{1}{n_1} + \frac{1}{n_2} ]}[/tex]
=> [tex]SE = \sqrt{ 0.0192(1- 0.0192 ) [ \frac{1}{700} + \frac{1}{2950} ]}[/tex]
=> [tex]SE = 0.00577[/tex]
Generally the margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} } * SE[/tex]
=> [tex]E = 1.96 * 0.00577[/tex]
=> [tex]E = 0.0113[/tex]
Generally 95% confidence interval is mathematically represented as
[tex](\^ p_1 - \^ p_2 )-E < p_1 - p_2 < \^( p_1 - \^ p_2) + E[/tex]
=> [tex](0.097 - 0.00237 )-0.0113< p_1 - p_2 < (0.097 - 0.00237 )+ 0.0113[/tex]
=> [tex]0.833< p_1 - p_2 < 0.1059[/tex]
