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MAT 135, Discussion Questions, 10.29
1. What kind of events could be modeled by a uniform distribution?
2. What are some properties of a normal distribution?
3. Give three examples of variables that are normally distributed.
4. What special properties does the standard normal distribution have (as compared with a
general normal distribution)?
5. How are z-scores related to the standard normal distribution?
6. How do we use the standard normal distribution to calculate probabilities?
7. Find the shaded area under the curve if the mean is 0 and the standard deviation is 1. The cutoff score here is 𝑧 = −1.28.
8. Find the shaded area under the curve for the cutoff value 𝑧 = 0.5.
9. Find the probability under the curve,
with mean 0 and standard deviation of 1.
10. Estimate the probability (area) under the normal
distribution curve for the z-scores shown on the
graph below.
11. Find the shaded area with a mean of 100, and a
standard deviation of 15.
12. Find the shaded area under the curve for the
cut-off values 𝑥 = 56, 𝑥 = 94 for the mean
of the distribution at 75, and a standard
deviation of 10.
13. Find the probability of the shaded
region with a mean of 20 and a standard
deviation of 3.5.
14. Mean is 21, and standard deviation is 5.
The cut-off values are 22 and 28.
15. Suppose that the mean IQ of a certain high
school is 109, with a standard deviation of
13. What is the standard score (z-score) of
a student with an IQ score of 125?
16. What is the probability that a student in the
school (above) will have an IQ below 125?
17. What is the probability that a student in the school (above) will have an IQ above 125?
18. What is the percentile ranking represented by an IQ of 125?
On the blank normal distribution graphs below, mark the mean and additional values in units of
standard deviations. Shade the region described by the situations for each problem.
19. The mean is 45, the standard deviation is 4. Find the probability of being above 49.
20. The mean is 62, the standard deviation is 13. Find the probability of being below 65.
21. The mean is 0.31, the standard deviation is 0.06. Find the probability of being between 0.22 and
0.48.
For each of the problems below, the percentage under the curve is indicated. Find the z or x cut-off
values for the region given the mean and the specified standard deviation.
22. Why is it important to be able to determine if data is normally distributed?
23. The area under the curve is 45% (graph on the left). With mean of 0 and the standard deviation
is 1.
24. Assume the mean is 0.88, and the standard deviation is 0.15.
25. Assume the mean is 0.72 and the standard deviation is 0.02.
26. The mean is 250, and the standard deviation is 5.
27. What is z if the shaded area is 75%?
What is the z score (same but for the sign
on both sides) if the probability of the
shaded region is 15%?
28. If a school wants to accept only the top 5% of SAT
and ACT scorers, what are the cut-off values for the
two tests? Recall that the SAT has a mean of 1498
and a standard deviation of 199 (total score), and
the ACT has a mean of 21 and a standard deviation
of 5.2. Draw a sketch of the normal distribution and
label it appropriately.
29. What conditions are necessary to approximate a
binomial distribution with a normal distribution?
30. When calculating the normal distribution data for the binomial distribution value for 𝑥 = 4, why
do we use bounds of 3.5 and 4.5 in the normal distribution?
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31. Approximate the binomial distribution for 𝑛 = 25, 𝑝 = 7 , 5 ≤ 𝑥 ≤ 7 using the normal
distribution. How close is the estimate?
32. What are the steps to calculating a normal probability plot?
33. When looking at a normal probability plot, what sort of things are you looking for?
34. Consider the normal probability plots below. Which ones appear sufficiently normal?
35. Use the graphs below of histograms and normal probability plots. Match the probability plot
with the histogram it goes with.
36. Use the images below to explain how the look of a boxplot translates to a normal probability
plot, and the skew of the graph.