For this problem, carry at least four digits after the decimal
in your calculations. Answers may vary slightly due to
rounding.
In a random sample of 63 professional actors, it was found that 36
were extroverts.
(a) Let p represent the proportion of all actors who
are extroverts. Find a point estimate for p. (Round your
answer to four decimal places.)
(b) Find a 95% confidence interval for p. (Round your
answers to two decimal places.)
lower limit | |
upper limit |
Give a brief interpretation of the meaning of the confidence interval you have found.
We are 95% confident that the true proportion of actors who are extroverts falls within this interval.
(c) Do you think the conditions np > 5 and nq > 5 are satisfied in this problem? Explain why this would be an important consideration.
Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately normal.
Solution :
Given that,
n = 63
x = 36
Point estimate = sample proportion = = x / n = 36/63=0.5714
1 - = 1- 0.5714 =0.4286
At 95% confidence level
= 1 - 95% = 1 - 0.99 = 0.01
/ 2 = 0.01 / 2 = 0.005
Z/2 = Z0.005 = 1.96 ( Using z table )
Margin of error = E = Z/2 * ((( * (1 - )) / n)
= 1.96 (((0.5714*0.4286) / 63)
E = 0.12
A 95% confidence interval for population proportion p is ,
- E < p < + E
0.5714-0.12 < p < 0.5714+0.12
0.4514< p < 0.6914
The 95% confidence interval for the population proportion p is : lower limit =0.45,upper limit = 0.69
(C)Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately normal.
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