Answer:
A. 99% CI for p: 0.125 < p < 0.259
B. 99% CI for p: 0.807 < p < 0.911
C. 95% CI for p: 0.776 < p < 0.844
D. 95% CI using t-distribution: 23.7 < µ < 40.3
Step-by-step explanation:
A.
[tex]p=\frac{x}{n} =\frac{44}{229} = 0.192139738[/tex]
Confidence = 99% = 0.99
[tex]\alpha[/tex]=1 - confidence = 1 - 0.99 = 0.01
Critical z-value =[tex]z_{\alpha/2}=z_{0.01/2}=z_{0.005}=2.58[/tex] (from z-table)
SE = Standard Error of p
[tex]SE=\sqrt{\frac{p(1-p)}{n} }[/tex]
[tex]SE=\sqrt{\frac{0.192139738(1-0.192139738)}{229} }[/tex]
[tex]SE=\sqrt{\frac{0.155222059}{229} }[/tex]
SE = 0.026035
E = Margin of Error
E = z-value * SE = 2.58 * 0.026035
E = 0.067170516
Lower Limit = p - E = 0.192139738 - 0.067170516 = 0.125
Upper Limit = p + E = 0.192139738 + 0.067170516 = 0.259
99% CI for p: 0.125 < p < 0.259
B.
[tex]p=\frac{x}{n} =\frac{256}{298} = 0.8590604[/tex]
Confidence = 99% = 0.99
[tex]\alpha[/tex]=1 - confidence = 1 - 0.99 = 0.01
Critical z-value =[tex]z_{\alpha/2}=z_{0.01/2}=z_{0.005}=2.58[/tex] (from z-table)
SE = Standard Error of p
[tex]SE=\sqrt{\frac{p(1-p)}{n} }[/tex]
[tex]SE=\sqrt{\frac{0.8590604(1-0.8590604)}{298} }[/tex]
[tex]SE=\sqrt{\frac{0.121075629}{298} }[/tex]
SE = 0.020157
E = Margin of Error
E = z-value * SE = 2.58 * 0.020157
E = 0.052004
Lower Limit = p - E = 0.8590604 - 0.052004 = 0.807
Upper Limit = p + E = 0.8590604 + 0.052004 = 0.911
99% CI for p: 0.125 < p < 0.259
C.
[tex]p=\frac{x}{n} =\frac{405}{500} = 0.81[/tex]
Confidence = 95% = 0.95
[tex]\alpha[/tex]=1 - confidence = 1 - 0.95 = 0.05
Critical z-value =[tex]z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96[/tex] (from z-table)
SE = Standard Error of p
[tex]SE=\sqrt{\frac{p(1-p)}{n} }[/tex]
[tex]SE=\sqrt{\frac{0.81(1-0.81)}{500} }[/tex]
[tex]SE=\sqrt{\frac{0.1539}{500} }[/tex]
SE = 0.017544
E = Margin of Error
E = z-value * SE = 1.96 * 0.017544
E = 0.034387
Lower Limit = p - E = 0.81 - 0.034387 = 0.776
Upper Limit = p + E = 0.81 + 0.034387 = 0.844
95% CI for p: 0.776 < p < 0.844
D.
x = sample mean = 32
s = sample standard deviation = 13
n = sample size = 12
Confidence = 95% = 0.95
[tex]\alpha[/tex]=1 - confidence = 1 - 0.95 = 0.05
Being the sample size less than 30, we use the statistical t.
df = degree of freedom = n - 1 = 12 - 1 = 11
Critical t-value =[tex]t_{\alpha/2,df}=t_{0.05/2,11}=t_{0.025,11}=2.201[/tex] (from t-table, two tails, df=11)
SE = standard error
[tex]SE=\frac{s_{x} }{\sqrt{n} } =\frac{13 }{\sqrt{12} }=3.752777[/tex]
E = margin of error
E = t-value * SE = 2.201 * 3.752777 = 8.259862
Lower Limit = x - E = 32 - 8.259862 = 23.7
Upper Limit = x + E = 32 + 8.259862 = 40.3
95% CI using t-distribution: 23.7 < µ < 40.3
Hope this helps!
