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 68 professional actors, it was found that 40 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.We are 5% confident that the true proportion of actors who are extroverts falls above this interval.    We are 95% confident that the true proportion of actors who are extroverts falls outside this interval.We are 5% 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.No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately binomial.    Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately binomial.No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately

Question
Asked Jul 29, 2019
725 views

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 68 professional actors, it was found that 40 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.We are 5% confident that the true proportion of actors who are extroverts falls above this interval.    We are 95% confident that the true proportion of actors who are extroverts falls outside this interval.We are 5% 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.No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately binomial.    Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately binomial.No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately 
check_circle

Expert Answer

Step 1

(a) Point estimation for p:

It is given that out of 68 professional actors, 40 are extroverts.

The point estimate (p-hat) is calculated as 0.5882, from the calculations given below.

help_outline

Image Transcriptionclose

40 68 = 0.5882 l

fullscreen
Step 2

(b) Confidence interval for p:

Here, the sample size, n = 68.

The probability of success p = 0.40

The probability of failure q = 0.60 (= 1 – 0.40)

The value of np = 27.2 (=68×0.40)

The value of nq = 40.8 (=68×0.60)

Since np and nq are greater than 5, p-hat is approximately normal.

The formula for confidence interval for the population proportion when np and nq are greater than 5 is as follows.

help_outline

Image Transcriptionclose

|p(1-p) Confidence interval = ptz2 n p(1-p) |p(1-p) p+z < a/2 a/2. where, p Sample proportion, Po Population proportion P(1-p) Z a2 Margin of error n P(1-p\=Lower limit of the mean p-Za2 |p(1-p) p+Za2 |= Upper limit of the mean n

fullscreen
Step 3

Computation of the 95% confidence interval for p:

Here, the sample proportion (p-hat) is 0.5882.

The confidence level is 0.95.

Thus, the level of significance, α is 0.05 (= 1 – 0.95).

For 95% confidence interval, the critical value for z0.025 ...

help_outline

Image Transcriptionclose

(0.5882 (1-0.5882) Lower Limit 0.5882 -1.96, 68 =0.5882-0.1170 [Rounded to the two decimal number] -0.47 Upper Limit 0.5882 +|1.96 /(0.5882(1-0.5882)) 68 =0.5882 0.1170 Rounded to the two decimal number] =0.71 95% confidence interval |p(1- p) ptZa2 -0.47< p<0.71 = (0.47,0.71)

fullscreen

Want to see the full answer?

See Solution

Check out a sample Q&A here.

Want to see this answer and more?

Solutions are written by subject experts who are available 24/7. Questions are typically answered within 1 hour.*

See Solution
*Response times may vary by subject and question.
Tagged in

Math

Probability

Other

Related Probability Q&A

Find answers to questions asked by student like you
Show more Q&A
add
question_answer

Q: . Initial diagnosis of a patient’s illness was potentially flu (F), cold (C), or hay fever (H), with...

A: For bayes' theorem, its always better to draw probabilities in tree diagram to visually analyse them...

question_answer

Q: Two fair dice are rolled at once. Find the conditional probability that the sum is 6, given that the...

A: Concept of conditional probability:For any two events A and B with P(B) &gt;0, the formula for condi...

question_answer

Q: 11.6.33 It is equally probable that the pointer on the spinner shown will land on any one of the eig...

A: It is possible for the pointer to land on a number that is both Odd and greater than 5. number  Odd ...

question_answer

Q: what is 63 million Americans under age 15 out of 314 million Americans in 2011?

A: Given, Americans under age 15 = 63 million               Total americans = 314 million

question_answer

Q: A pair of dice is rolled, and the larger of the two numbers showing is recorded.  Find the expecrted...

A: Given that a pair of dice is rolled and the larger of the two numbers is noted.Total number of possi...

question_answer

Q: P(56&lt;X&lt;68) assume that the random variable X is normally distributed with mean is 50 and stand...

A: The normal distribution:A continuous random variable X is said to follow normal distribution if the ...

question_answer

Q: Let Xi be IID random variables which have the same law as X. Let L(t) = E(e^tX.) Suppose that this i...

A: Moment generating function (MGF):The moment generating function (MGF) of a random variable X, denote...

question_answer

Q: Help me with a question on random variable.Question: For a random variable X, its probability densit...

A: The mean(mX) of a random variable X is given as:

question_answer

Q: Find z such that 60% of the standard normal curve lies to the left of z. (Round your answer to two d...

A: It is required to find the z-value such that, P (Z &lt; z) = 0.60.