a random sample of adults was surveyed about their exercise habits. of the 100 surveyed, 56 stated they…
a random sample of adults was surveyed about their exercise habits. of the 100 surveyed, 56 stated they exercise regularly. which of the following is the 95% confidence interval for p, the proportion of all adults who exercise regularly? find the z - table here. (0.50, 0.62) (0.48, 0.64) (0.46, 0.66) (0.43, 0.69)
Answer
Explanation:
Step1: Calculate sample proportion
The sample proportion $\hat{p}=\frac{56}{100} = 0.56$.
Step2: Find z - score
For a 95% confidence interval, the z - score $z_{\alpha/2}=1.96$.
Step3: Calculate standard error
The standard error $SE=\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=\sqrt{\frac{0.56\times(1 - 0.56)}{100}}=\sqrt{\frac{0.56\times0.44}{100}}=\sqrt{0.002464}\approx0.0496$.
Step4: Calculate margin of error
The margin of error $ME = z_{\alpha/2}\times SE=1.96\times0.0496\approx0.0972$.
Step5: Calculate confidence interval
The lower limit is $\hat{p}-ME=0.56 - 0.0972=0.4628$ and the upper limit is $\hat{p}+ME=0.56+0.0972 = 0.6572$.
Answer:
$(0.46,0.66)$