Sampling Distributions for Means

Worksheet 2

Open the interactive version of the normal tables. (Opens a separate browser window.)

Obtain some pictures of normal curves. Print them out and use them to help solve problems.

Note: The typical symbol for the sample mean, an X with an overbar, doesn't integrate well into web documents. Therefore, I've merely refered to it as either "X-bar" (what it's often called, anyway) or "the sample mean" (what it is).


1. A jar of Jiffy Peanut Butter (labeled as a 32 oz. jar) is selected randomly off the end of the assembly line. The weight of the jar is measured. If X measures this weight then X has the N(32.3, 0.40) distribution (the filling machine is calibrated this way). That is, the mean fill is 32.3 oz., the standard deviation of the fills is 0.40 oz..

a) Identify the response variable being measured. Is it categorical or quantitative?
b) Identify the population and parameters.
c) What is the average fill of all Jiffy jars? Is this value a statistic?
d) What's the probability a randomly selected jar is filled with less than 32.0 fl oz? Such a can would be "under-volume."
e) What proportion of all Jiffy jars are under-volume?
f) Consider a simple random sample of 6 jars. Let X-bar be the sample mean weight for these 6 jars. Is X-bar a parameter or a statistic?
g) What are the mean and standard deviation of the (sampling) distribution for the sample mean?
h) What is the shape of the sample means's distribution?
i) Find the probability the sample mean fill of the six jars is less than 32.0, P[ X-bar < 32.0].
j) Now take a simple random sample of 24 jars. (The sample size has been multiplied by 4.) What are the mean and standard deviation of the distribution for X-bar? How has the standard deviation of changed with the quadrupling of the sample size?
k) Find the probability the sample mean fill of the 24 jars is less than 32.0.

2. The weights, in pounds, of Portuguese Water Dogs has the N(45, 3) distribution.

a) If one dog is selected at random and X is its weight, find the probability X is within 1 pound of the mean weight of 45, P[44 < X < 46].
b) Draw a simple random sample of 40 dogs. Find the probability the sample mean weight is within 1 pound of the mean weight of 45.
c) Draw a simple random sample of 80 dogs. Find the probability the sample mean weight is within 1 pound of the mean weight.

3. The number of murders in Oswego County in a year is a random variable. Suppose the mean of the distribution of yearly murders is 1.8 and the standard deviation is 1.5.

a) Let X represent the number of murders in some (past, present or future) year. Does X have a normal distribution? If not, what distinguishes the distribution of X from a normal distribution? (Skew? Discreteness?)
b) Suppose we are interested in predicting the number of murders in future years (jail cells for these dangerous criminals must be built in advance). Take for instance the next five years. Let X-bar be the average yearly figure for the next five years (assume the values for the individual years are independent). What are the mean and standard deviation of X-bar? (The sample size and the hypothesized shape of the distribution for X lead me to conclude that the Central Limit Theorem should not be applied in order to approximate probabilities involving for n = 5.)
c) Suppose instead n = 25. Using the Central Limit Theorem, compute the approximate probability that the sample mean number of murders for the 25 years is above 2.5. (To solve the problem first find the mean and standard deviation of the sampling distribution for when n = 25.)
d) If the sample mean number of murders over the next 25 years exceeds 2.5, what can you say about the total number of murders over the same period?

4. Stock market analysts measure the risk of a given company's stock using an quantity called the "b index." (How this value is computed is not important-it just represents one way stock analysts measure a stock's risk.) A company with b > 1 has stock with above average investment risk; b < 1 means it has below average risk (its a relatively safe investment). The distribution of this variable (for companies listed on the New York Stock Exchange) has mean 1 and standard deviation 0.26. 0 is the lowest possible risk (a few companies-very established, consistent performers-have b very near 0); technically a risk can be as high as possible- in reality the maximum is near 5.

a) Identify the response variable being measured. Is it categorical or quantitative?
b) Identify the population and all parameters.
c) How can the average risk be 1 when the minimum is 0 and the maximum is 5?
d) Take a simple random sample of one stock and measure its b index. With the information given is it possible to compute P[X > 1.15]? Why (not)? If not, how can this probability be found?
e) Suppose 20 stocks are selected by simple random sample from all listed stocks. Let X-bar be the average risk for the 20. Would you consider X-bar a parameter or a statistic?
f) What are the mean and standard deviation of X-bar's distribution?
g) Use the Central Limit Theorem (assume n = 20 is sufficiently large) to approximate P[X-bar > 1.15].
h) My stock portfolio contains 20 stocks with (sample) mean b index of 1.151. Using the previous part's answer, what can you say about my portfolio?