Sampling Distributions
1 Suppose the amount students at PSU spent on textbooks this semester is a normal random variable with mean μ= $360 and standard deviation σ = $90.
a. Use the empirical rule for bell-shaped data to determine intervals that will contain about 68%, 95% and 99.7% of amounts spent on textbooks by PSU students. Show work for each interval.
b. With Software, find the cumulative probability for $300. Note: The notation for the probability you’re finding is P(X ≤ $300).
c. Write a sentence interpreting the value found in the previous part that would be understood by somebody with no training in statistics.
d. What proportion of students spent more than $535? (Start like you did for part b, but then you’ll have to do a “by hand” calculation using the complement rule to determine the final answer. [Note: The notation for the probability you’re finding is P(X > $535).]
e. What is the probability that a randomly selected student will have spent between $300 and $535? Show work. [Note: You’ll be able to utilize information from parts b and d.]
f. Calculate a z-score for $400.
g. Use the Standard Normal Table in the text (or that linked to in the notes) to determine the cumulative probability for the z-score you calculated in the previous part.
h. Write a sentence interpreting the value found in the previous part that would be understood by somebody with no training in statistics.
i. Use Software to find the probability a randomly selected student spent less than $400. If you didn’t get the same answer as you did in part g, figure out why. The answers should be the same.
j. What proportion of students spent more than $400 on textbooks? Show any work.
k. Find the probability an amount spent is less than or equal to $240. Solve the problem using Standard Normal Table. Show calculation of the z-score as part of your solution. (You can verify your answer using Software if you wish.)
l. With Software, find the 75th percentile of amounts spent on textbooks. This will involve finding the Inverse Cumulative Probability!
m. Write a sentence interpreting the value found in the previous part that would be understood by somebody with no training in statistics.
n. In the Standard Normal Table search for the cumulative probability most near 0.75. You’re looking “inside” the table where the probabilities are for the value closest to 0.7500. What is the z-score with this cumulative probability?
o. Refer to the previous part. Determine the amount spent on textbooks having the z-score found in the previous part. Show work. [Note: Compare your answer to the answer found in part l. If the answers differ by a lot, something’s not right.]
2 The term sampling frame refers to the group that actually had a chance to get into the sample. Ideally, this is the same as the population of interest, but sometimes it isn’t. In the following situation, describe the population, the sampling frame, the sample, the parameter of interest, and the statistic.
A Gallup Poll is done using random digit dialing to reach individuals in households with land-line telephones. The purpose is to estimate the proportion of U.S. adults who favor stronger gun control laws. One-thousand persons are sampled, and 63% favor stronger gun control.
a. Population =
b. Sampling frame =
c. Parameter =
d. Sample =
e. Statistic =
3 To get a better understanding of the Central Limit Theorem as discussed in the lecture notes you can visit and review a simulation program a (this program is also the one in the lecture notes):
http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html
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