# Population and Standard Mean

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1. Assume the population standard deviation is a=25. (The a is actually the funny looking 0 with the line that goes from the top right up at an angle) Compute the standard error of the mean O (funny o with line) line over x, for sample sizes of 50, 100, 150, and 200. What can you say about the size of the standard error of the mean as the sample size is increased?

2. The College Board American College Testing Program reported a population mean Sat score of (funny looking) u = 1020. Assume that the population standard deviation is funny looking o with line top right = 100,

a. What is the probability that a random sample of 75 students will provide a sample mean score within 20 of the population mean?

b. What is the probability a random sample of 75 students will provide a sample mean Sat score within 20 of the population mean?

3. The average annual cost of automobile insurance is $687. Use this value as the population mean and assume that the population standard deviation is funny looking o with line on top =$230. Consider a sample of 45 automobile insurance policies.

a. Show the sampling distribution of line over x where line over x is the sample mean annual cost of automobile insurance.

b. What is the probability that the sample mean is within $100 of the population mean?

c. What is the probability that the sample mean is within $25 of the population mean?

d. What would you recommend if an insurance agency wanted the sample mean to estimate the population mean within + or - $25?

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##### Solution Summary

Populations and standard means are examined. Probabilities for a random sample SAT score are provided.

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Please see the attached file.

1. Solution. We use a formula for the standard error . In this question, .

1) When n=50,

2) When n=100,

3) When n=150,

4) When n=200,

From the above calculations and results, we can say that the standard error would decrease as the sample size is increased.

2. Solution. In this question, .

a) The sample size n=75, so the standard error of the mean is

By the Central ...

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

###### Recent Feedback

- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
- "Thank you"
- "Thank you very much for your valuable time and assistance!"

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