Respuesta :
Answer: a. A point estimate for p is 0.597 .
b. The 95% confidence interval for p.
Lower limit = 0.48
Upper limit = 0.71
Step-by-step explanation:
Given : Sample size of professional actors : n= 67
Number of extroverts : x= 40
Let p represent the proportion of all actors who are extroverts.
a. The point estimate for p = sample proportion = [tex]\hat{p}=\dfrac{x}{n}[/tex]
[tex]=\dfrac{40}{67}=0.597[/tex]
b. Confidence interval for population proportion :
[tex]\hat{p}\pm z^*\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]
Since the critical value for 95% confidence interval is 1.96 , so the 95% confidence interval for p would be
[tex]0.597\pm (1.96)\sqrt{\dfrac{0.597(1-0.597)}{67}}[/tex]
[tex]0.597\pm (1.96)\sqrt{0.0036}[/tex]
[tex]0.597\pm (1.96)(0.06)[/tex]
[tex]0.597\pm 0.1176[/tex]
[tex](0.597-0.1176,\ 0.597+0.1176)\\\\=(0.4794,\ 0.7146)\\\\\approx(0.48,\ 0.71)[/tex]
In the 95% confidence interval for p.
Lower limit = 0.48
Upper limit = 0.71
Answer: the upper limit is 0.58
The lower limit is 0.56
Step-by-step explanation:
Total number of professional actors that was sampled is 67. it was found that 40 were extroverts.
a) the proportion of all actors who are extroverts would be
p = 40/70 = 0.5714
Therefore, the proportion of all actors who are not extroverts would be
q = 1 - p = 1 - 0.5714 = 0.4286
b) The formula for determining the confidence interval for a population proportion is expressed as
p ± z[p(1 - p)/n
The z value for a 95% confidence level is 1.96.
Therefore
Confidence interval
= 0.5714 ± 1.96[0.5714(1 - 0.5714)/67
= 0.5714 ± 1.96(0.2449/67)
= 0.5714 ± 0.00716
The upper limit would be
0.5714 + 0.00716 = 0.58
The lower limit would be
0.5714 - 0.00716 = 0.56
