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Problems involving x-bar, Schemes and Mind Maps of Probability and Statistics

Rice UniversityProbability and Statistics

At a certain bank, the mean checking account balance is =$950 and the population standard deviation is =$104. (Assume the distribution is normal).

Typology: Schemes and Mind Maps

2022/2023

Uploaded on 02/28/2023

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Answers
1. A sample of size n=400 is randomly selected from a non-normal population with
a mean =240 and standard deviation =40.
a. Find
)( 235XP
b. Find
)( 244236 XP
c. Find
)( 245XP
2. At a certain bank, the mean checking account balance is =$950 and the
population standard deviation is =$104. (Assume the distribution is normal).
a. What is the probability that the sample mean of 256 randomly chosen
checking account balances is between $944 and $956?
b. What is the probability that a randomly selected individual checking account
is between $944 and $956?
c. What is the probability that the sample mean of 256 randomly chosen
checking account balances is between $940 and $960?
d. What is the probability that a randomly selected individual checking account
is between $940 and $960?
3. Assume the average city in the US has 45,200 residents and the population
standard deviation is 33,800. Suppose you randomly sample 49 American cities.
a. Describe the population you are sampling from. Do you think the distribution
of city sizes would be normal?
b. Calculate the probability the sample mean city size is greater than 52,000.
c. Calculate the probability the sample mean city size is less than 48,000.
d. Suppose that instead of randomly sampling across all American cities, you
purposely selected only among cities in California. How would this change
the probabilities calculated in parts b and c?
e. Given the information we have, explain why we cannot calculate the
probability that an individual randomly selected city would be within a certain
size range.
4. In 2002, the population mean yearly tuition at state universities in the US was
$4,260. Assume the population standard deviation is $900.
a. Calculate the probability that a sample of 36 state universities has a mean
tuition within $200 of the population mean.
b. Calculate the probability that a sample of 100 state universities has a mean
tuition within $200 of the population mean.
c. Explain why the probability calculated in part b is larger than the probability
calculated in part a.
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X Answers

  1. A sample of size n=400 is randomly selected from a non-normal population with a mean =240 and standard deviation =40. a. Find P ( X 235 ) b. Find P ( 236 X 244 ) c. Find P ( X 245 )
  2. At a certain bank, the mean checking account balance is =$950 and the population standard deviation is =$104. (Assume the distribution is normal). a. What is the probability that the sample mean of 256 randomly chosen checking account balances is between $944 and $956? b. What is the probability that a randomly selected individual checking account is between $944 and $956? c. What is the probability that the sample mean of 256 randomly chosen checking account balances is between $940 and $960? d. What is the probability that a randomly selected individual checking account is between $940 and $960?
  3. Assume the average city in the US has 45,200 residents and the population standard deviation is 33,800. Suppose you randomly sample 49 American cities. a. Describe the population you are sampling from. Do you think the distribution of city sizes would be normal? b. Calculate the probability the sample mean city size is greater than 52,000. c. Calculate the probability the sample mean city size is less than 48,000. d. Suppose that instead of randomly sampling across all American cities, you purposely selected only among cities in California. How would this change the probabilities calculated in parts b and c? e. Given the information we have, explain why we cannot calculate the probability that an individual randomly selected city would be within a certain size range.
  4. In 2002, the population mean yearly tuition at state universities in the US was $4,260. Assume the population standard deviation is $900. a. Calculate the probability that a sample of 36 state universities has a mean tuition within $200 of the population mean. b. Calculate the probability that a sample of 100 state universities has a mean tuition within $200 of the population mean. c. Explain why the probability calculated in part b is larger than the probability calculated in part a.
  1. Suppose the population mean number of books at non-academic libraries in the US is 35,500 books with a population standard deviation of 15,000 books. Assume you randomly sample 64 libraries. a. Describe the population of the problem. b. Calculate the probability of obtaining a sample mean number of books greater than 40,000. c. Calculate the probability of obtaining a sample mean number of books greater than 33,625. d. P ( X >^50 ,^000 )≈^0. Verify that this is true and explain what it means. Does this imply that there is no public library in the US with more than 50, books? e. Explain why the population of library sizes (in terms of number of books) is probably not normally distributed.

e. Given the information we have, explain why we cannot calculate the

probability that an individual randomly selected city would be within a

certain size range.

Because the distribution is not normal. We don’t know what the

distribution of the population of city sizes is.

4. In 2002, the population mean yearly tuition at state universities in the US

was $4,260. Assume the population standard deviation is $900.

a. Calculate the probability that a sample of 36 state universities has a

mean tuition within $200 of the population mean..

b. Calculate the probability that a sample of 100 state universities has a

mean tuition within $200 of the population mean..

c. Explain why the probability calculated in part b is larger than the

probability calculated in part a.

The larger sample decreases the standard deviation of X.

5. Suppose the population mean number of books at non-academic libraries in

the US is 35,500 books with a population standard deviation of 15,

books. Assume you randomly sample 64 libraries.

a. Describe the population of the problem.

All non-academic libraries in the US.

b. Calculate the probability of obtaining a sample mean number of books

greater than 40,000.

c. Calculate the probability of obtaining a sample mean number of books

greater than 33,625.

d. P ( X >^50 ,^000 )≈^0. Verify that this is true and explain what it means.

Does this imply that there is no public library in the US with more than

50,000 books?

P ( X >50,000)= P ( Z >7.73)≈

No, this is only saying that the chance that a sample mean is greater than

50,000 is approximately zero. The chance that an individual library has

over 50,000 books is P(X>50,000), which equals .166.

e. Explain why the population of library sizes (in terms of number of

books) is probably not normally distributed.

The distribution of library sizes probably has the same general shape as the

distribution of city sizes. The outlier cities on the right side of the

distribution probably have large central libraries that would be outliers in

the distribution of libraries.