Question

In: Statistics and Probability

A simple random sample of size nequals60 is obtained from a population with muequals69 and sigmaequals5....

A simple random sample of size nequals60 is obtained from a population with muequals69 and sigmaequals5. Does the population need to be normally distributed for the sampling distribution of x overbar to be approximately normally​ distributed? Why? What is the sampling distribution of x overbar​? Does the population need to be normally distributed for the sampling distribution of x overbar to be approximately normally​ distributed? Why? A. No because the Central Limit Theorem states that only if the shape of the underlying population is normal or uniform does the sampling distribution of x overbar become approximately normal as the sample​ size, n, increases. B. Yes because the Central Limit Theorem states that the sampling variability of nonnormal populations will increase as the sample size increases. C. Yes because the Central Limit Theorem states that only for underlying populations that are normal is the shape of the sampling distribution of x overbar ​normal, regardless of the sample​ size, n. D. No because the Central Limit Theorem states that regardless of the shape of the underlying​ population, the sampling distribution of x overbar becomes approximately normal as the sample​ size, n, increases. Your answer is correct. What is the sampling distribution of x overbar​? Select the correct choice below and fill in the answer boxes within your choice. ​(Type integers or decimals rounded to three decimal places as​ needed.) A. The sampling distribution of x overbar is skewed left with mu Subscript x overbarequals nothing and sigma Subscript x overbarequals nothing. B. The sampling distribution of x overbar follows​ Student's t-distribution with mu Subscript x overbarequals nothing and sigma Subscript x overbarequals nothing. C. The sampling distribution of x overbar is normal or approximately normal with mu Subscript x overbarequals nothing and sigma Subscript x overbarequals nothing. D. The sampling distribution of x overbar is uniform with mu Subscript x overbarequals nothing and sigma Subscript x overbarequals nothing.

Solutions

Expert Solution

The mean of the population is and the standard deviation is

A sample of size n=60 is drawn from this population.

Here the sample size is greater than 30 and hence the central limit theorem applies. That is as per the central limit theorem, the distribution of sample mean is approximately normal, irrespective of the distribution of the population.

Does the population need to be normally distributed for the sampling distribution of to be approximately normally​ distributed?

ans: D. No because the Central Limit Theorem states that regardless of the shape of the underlying​ population, the sampling distribution of becomes approximately normal as the sample​ size, n, increases.

What is the sampling distribution of ?

has normal distribution with mean and standard deviation ( or called standard error of mean)

ans: C. The sampling distribution of is normal or approximately normal with and


Related Solutions

Suppose a simple random sample of size nequals200 is obtained from a population whose size is...
Suppose a simple random sample of size nequals200 is obtained from a population whose size is Upper N equals 15 comma 000 and whose population proportion with a specified characteristic is p equals 0.6 . ​(a) Describe the sampling distribution of ModifyingAbove p with caret. Choose the phrase that best describes the shape of the sampling distribution below. A. Approximately normal because n less than or equals 0.05 Upper N and np left parenthesis 1 minus p right parenthesis less...
A simple random sample of size nequals40 is obtained from a population with muequals66 and sigmaequals14....
A simple random sample of size nequals40 is obtained from a population with muequals66 and sigmaequals14. ​(a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample​ mean? Assuming that this condition is​ true, describe the sampling distribution of x overbar. ​(b) Assuming the normal model can be​ used, determine ​P(x overbarless than69.6​). ​(c) Assuming the normal model can be​ used, determine ​P(x overbargreater than or equals67.8​).
A simple random sample of size n = 15 is obtained from a population with μ...
A simple random sample of size n = 15 is obtained from a population with μ = 97 and σ = 23. Enter your answer as an area under the curve with 4 decimal places. P(x⎯⎯⎯ ≤ 94)
A simple random sample of size nequals37 is obtained from a population with muequals61 and sigmaequals14....
A simple random sample of size nequals37 is obtained from a population with muequals61 and sigmaequals14. ?(a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample? mean? Assuming that this condition is? true, describe the sampling distribution of x overbar. ?(b) Assuming the normal model can be? used, determine ?P(x overbarless than65.1?). ?(c) Assuming the normal model can be? used, determine ?P(x overbargreater than or equals62.4?).
A simple random sample of size nequals47 is obtained from a population with muequals67 and sigmaequals16....
A simple random sample of size nequals47 is obtained from a population with muequals67 and sigmaequals16. ?(a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample? mean? Assuming that this condition is? true, describe the sampling distribution of x overbar. ?(b) Assuming the normal model can be? used, determine ?P(x overbarless than70.9?). ?(c) Assuming the normal model can be? used, determine ?P(x overbargreater than or equals68.9?).
A simple random sample of size n = 64 is obtained from a population with µ...
A simple random sample of size n = 64 is obtained from a population with µ = 112 and σ = 24. What is P(x ¯ ≤ 109.5)? (Give answer in decimal form and round answer to two decimal places)
Suppose a simple random sample of size n=75 is obtained from a population whose size is...
Suppose a simple random sample of size n=75 is obtained from a population whose size is N= 30,000 and whose population proportion with a specified characteristic is p= 0.4 . A) Determine the standard deviation of the sampling distribution of p hat (Round to 6 decimals) B) What is the probability of obtaining x=33 or more individuals with the​ characteristic? That​ is, what is ​P(p ≥0.44​)? (Round to 4 decimals)
Suppose a simple random sample of size n=1000 is obtained from a population whose size is...
Suppose a simple random sample of size n=1000 is obtained from a population whose size is N=2,000,000 and whose population proportion with a specified characteristic is p = 0.22 a. What is the probability of obtaining x=250 or more individuals with the​ characteristic?
Suppose a simple random sample of size nequals=7575 is obtained from a population whose size is...
Suppose a simple random sample of size nequals=7575 is obtained from a population whose size is Upper N equals 30 comma 000N=30,000 and whose population proportion with a specified characteristic is p equals 0.4 .p=0.4. Complete parts ​(a) through​ (c) below. ​(a) Describe the sampling distribution of ModifyingAbove p with caretp. Choose the phrase that best describes the shape of the sampling distribution below. A. Not normal because n less than or equals 0.05 Upper Nn≤0.05N and np left parenthesis...
Suppose a simple random sample of size n=75 is obtained from a population whose size is...
Suppose a simple random sample of size n=75 is obtained from a population whose size is N=15,000 and whose population proportion with a specified characteristic is p=0.8. a) Determine the mean of the sampling distribution of p with caret. Determine the standard deviation of the sampling distribution of p with caret. b) What is the probability of obtaining x=66 or more individuals with the​ characteristic? That​ is, what is ​P(p with caret greater than or equals 0.88)? What is the...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT