Question

In: Statistics and Probability

After numerous studies, it is estimated that the age of college students follows a normal distribution...

After numerous studies, it is estimated that the age of college students follows a normal distribution with a mean of 20.7 years and a standard deviation of 1.8 years. A random sample of 16 college students is taken and the sample mean and sample standard deviation are found to be 21.5 years and 2.5 years, respectively.

a. Is the sample average of 21.5 considered an outlier (in other words, what is the chance of getting a sample mean at least as large as 21.5)?

b. The calculated value of the sample standard deviation is larger than σ, is 2.5 years enough larger than 1.8 that the estimate of σ does not seem reasonable?

c. No matter what you got for an answer in part b, assume we no longer trust σ = 1.8 to be valid. Calculate the same probability as you did in part a, but this time, assume that σ is unknown.

d. Is there a significant difference in parts A and C?

Expert Solution

Solution:

a. Is the sample average of 21.5 considered an outlier (in other words, what is the chance of getting a sample mean at least as large as 21.5)?

Answer: It is required to find:

Using the z-score formula, we have:

Now using the standard normal table, we have:

Since the probability is less than 0.05, therefore, the sample average of 21.5 can be considered an outlier.

b. The calculated value of the sample standard deviation is larger than σ, is 2.5 years enough larger than 1.8 that the estimate of σ does not seem reasonable?

Answer: The null and alternative hypotheses are:

Under the null hypothesis, the test statistic is:

The p-value is:

Since the p-value is less than the significance level, we, therefore, reject the null hypothesis and conclude that the estimate of σ is not reasonable

c. No matter what you got for an answer in part b, assume we no longer trust σ = 1.8 to be valid. Calculate the same probability as you did in part a, but this time, assume that σ is unknown.

Answer: It is required to find:

Using the t-score formula, we have:

Now using the standard normal table, we have:

Since the probability is greater than 0.05, therefore, the sample average of 21.5 cannot be considered an outlier.

d. Is there a significant difference in parts A and C?

Answer: Yes, there is a significant difference in parts A and C.

orchestra answered 1 year ago

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