Question

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 60 professional actors, it was found that 35 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 all actors who are extroverts falls outside of 0.46 and 0.71.We are 5% confident that the true proportion of all actors who are extroverts falls between 0.46 and 0.71.    We are 5% confident that the true proportion of all actors who are extroverts falls above 0.46 and 0.71.We are 95% confident that the proportion of all actors who are extroverts falls between 0.46 and 0.71.

(c) Do you think the conditions np is greater than or equal to 15 and n*(1 - p) is greater than or euqal to 15 are satisfied in this problem? Explain why this would be an important consideration.

A No, the conditions are not satisfied. This is important because it allows us to say that the sampling distribution of p̂ is approximately normal.

B Yes, the conditions are satisfied. This is important because it allows us to say that the sampling distribution of p̂ is approximately binomial.

C No, the conditions are not satisfied. This is important because it allows us to say that the sampling distribution of p̂ is skewed right.

D Yes, the conditions are satisfied. This is important because it allows us to say that the sampling distribution of p̂ is approximately normal.

Answer 1

In a random sample of 60 professional actors, it was found that 35 were extroverts.

This is binomial experiment since there are only two outcomes (introvert or extrovert) and there are many independent trials with the same probability of success (being an extrovert).

(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.)

Let 'X' be the no. of introverts . Therefore

Where the sample proportion or the point estimate

(b) Find a 95% confidence interval for p. (Round your answers to two decimal places.)

The 95% confidence interval for population proportion is

Where

Critical value   Found using normal percentage tables, excel func 'normsinv(0.025)' we have to take positive value

Standard error =

=

=

Substituting all the values

lower limit	0.4586
upper limit	0.7081

Confidence interval gives a range for the true parameter with a possibility of error or (1-) certainty. Another explanation is that if carry the same test with different samples then there is (1-) of the ranges will have the true parameter.Therefore

Give a brief interpretation of the meaning of the confidence interval you have found.

We are 95% confident that the true proportion of all actors who are extroverts falls outside of 0.46 and 0.71.

We are 5% confident that the true proportion of all actors who are extroverts falls between 0.46 and 0.71.    

We are 5% confident that the true proportion of all actors who are extroverts falls above 0.46 and 0.71.

We are 95% confident that the proportion of all actors who are extroverts falls between 0.46 and 0.71.

(c) Do you think the conditions np is greater than or equal to 15 and n*(1 - p) is greater than or euqal to 15 are satisfied in this problem? Explain why this would be an important consideration.

As calculated above, we can see we approximated to normal distribution (critical value). There are certain conditions to apply the approximation.

np > 15   We have np = 60 * 0.5833 = 35 > 15

or

n(1-p) > 15   We have np(1-) = 60 * (1-0.5833) = 25 > 15

A No, the conditions are not satisfied. This is important because it allows us to say that the sampling distribution of p̂ is approximately normal.

B Yes, the conditions are satisfied. This is important because it allows us to say that the sampling distribution of p̂ is approximately binomial.

C No, the conditions are not satisfied. This is important because it allows us to say that the sampling distribution of p̂ is skewed right.

D Yes, the conditions are satisfied. This is important because it allows us to say that the sampling distribution of p̂ is approximately normal.