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Week 4 Homework Hints
Chapters 7, 8
Problem 7.32 General Background For Understanding This Problem:
Note: this chapter’s z-score formula differs from the earlier z-score formula on p. 227,
formula 6.2, because this chapter is all about finding the probability that a sample MEAN
(“X-bar”) differs from a population mean (U). The formula on p.227 finds the
probability that an INDIVIDUAL VALUE (X) differs from a population or sample
mean.)
Now, on p. 271, the authors state that the distance a given sample mean is from the
center of the distribution can be determined using the z-value for the sampling
distribution of the sample mean: (formula 7.4)
We can use this to solve probability problems involving sample means.
One other twist may be necessary: if the sample that you select is relatively large
compared to the population – that is, if n > 5% of N – then you can actually reduce the
standard error of the mean using the finite population correction factor; in this case, the
formula becomes: (formula 7.5, p. 271)
Specific Hints For Solving Problem 7.32:
Problem 7.32, a,b,c,:
For this problem, you will want to use formula 7.5 on p. 271. Part a says “determine the
probability that the random sample of 40 will have a sales average less than $400,000.
So we know that “X-bar”, the sample mean, is 400,000. We also know that n, the sample
size = 40. We need to know what u, the population mean is, and that’s given in the
problem (“last year, the overall company average was $417,330”). Now we also need the
“finite population correction factor” since the population size (N) is a sales force of 220.
The sample size (n) as we already know is 40. The fpc is used when your sample size is
greater than 5% of your population size. So n/N = 40/220 = .18,or 18%, so we need to
use the fpc. The population standard deviation (sigma) is also given to you. You should
now have everything you need to plug into the equation and calculate your z-score. Once
you calculate the z-score, you need to go to the normal table and look up the appropriate
area under the curve (or use the Excel function NORM.S.DIST, plug in your z-score, put
“TRUE” in the “Cumulative” box). Remember, your z-score will be negative, and you’re
looking for the area under the curve (the probability) of getting a sales average LESS
than $400,000….so you’re looking for the area under the curve to the left of that value.
For 7.32 b, review the discussion of the Central Limit Theorem in the chapter (see
Theorem 3 on p. 268).
For 7.32c, the “standard deviation of the distribution of sample means” is also called the
“standard error” (see Theorem 2 on p.267). Hint: this is equivalent to the denominator of
the formula you used to solve problem 7.32a.
Problem 7-54, to be clear, requires a probability calculation, even though the last line in
the problem states “Discuss how they could use this information…..”. We want to use
the formula for proportions, just like we used a formula for sample means in the prior
problem. The formula to be used when trying to find the probability of getting a sample
proportion of a certain size, when you know the population proportion and sample size, is
shown below…see also the examples on pp.282-285; here, p is the same as p-bar (sample
proportion), and p in the formula below is the same as the “pie” symbol,, which is the
population proportion. In this problem, the managers tested a simple random sample of n
= 200 sprinkler valves from the population and found x = 190 defect-free valves. So the
sample proportion is:
x 190
p 
 0.95
n 200
This is less than the required 0.97 for the population, which is your P.
z
pp

σp
pp
p(1  p)
n
Once you get your calculated z score from the formula above, determine the probability
of getting that z-score (or less) by using the Excel function Norm.S. Dist and plugging in
your z score, putting “TRUE” in the cumulative box:
Problem 8-12:
For this problem, pay particular attention to pages 304-310. A good example to follow is
Example 8-3 on p. 309. You can use the Excel function “Confidence.T” to determine the
margin of error. To create the actual confidence interval, you subtract the margin of error
from the mean to get the lower end, and then add the margin of error to the mean to arrive
at the upper end. Of course, you will need to calculate the mean and standard deviation
once you have your data in a single column. Please remember that you do not have to
construct a box and whisker plot that is mentioned in part c of the problem. To use the
Confidence.T function, you need to input:
 the alpha level (so, for example a 90% confidence level would require an alpha of
.10, a 99% confidence level would require and alpha of .01, etc.)
 the standard deviation
 the sample size
Problem 8-38 part a only:
Part a says “A previous study indicated that….for (all) long distance calls”, so assume
we know the population standard deviation, or sigma. You can use the formula on p. 315
(equation 8.6) to manually calculate this, or you can simply create the formula in Excel
by plugging in the values for z, sigma, and e, then squaring.
The formula to use is shown below:
Problem 8-56:
The formula for the confidence interval around a proportion is covered on p. 322, formula
8.10. You can do this manually, or create a simple Excel formula to plug in the values.
The actual formula to use is listed on the slide immediately below; an example of how to
calculate this can be found in the second slide: