1. Consider the uniform distribution shown below. Find the probability that x is greater
than 6. Discuss the relationship between area under a density curve and probability.
2. Use the given degree of confidence and sample data to construct a confidence interval
for the population mean. Assume that the population has a normal distribution.
A laboratory tested twelve chicken eggs and found that the mean amount of cholesterol
was 204 milligrams with s = 13.3 milligrams. Construct a 95 percent confidence
interval for the true mean cholesterol content of all such eggs.
3. Thirty randomly selected students took the calculus final. If the sample mean was 77
and the standard deviation was 5.6, construct a 99 percent confidence interval for the
mean score of all students. Assume that the population is normally distributed.
4. The principal randomly selected six students to take an aptitude test. Their scores
were:71.9, 72.7, 88.4, 77.8, 87.3, 79.5
a) Determine a 90 percent confidence interval for the mean score for all students.
5. A survey of 865 voters in one state reveals that 408 favor approval of an issue before
the legislature. Construct the 95% confidence interval for the true proportion of all
voters in the state who favor approval.
6. When 283 college students are randomly selected and surveyed, it is found that 100
own a car. Find a 99% confidence interval for the true proportion of all college
students who own a car.
7. Of 92 adults selected randomly from one town, 68 have health insurance. Find a 90%
confidence interval for the true proportion of all adults in the town who have health
insurance.
8. Identify the correct distribution (z, t, or neither) for each of the following.
Sample Size / Standard Deviation / Shape of Distribution / Z or T or Neithern = 35 / s = 4.5 / Somewhat Skewed
n = 20 / s = 4.5 / Bell Shaped
n = 25 / = 4.5 / Bell Shaped
n = 20 / = 4.5 / Extremely Skewed
9. Find the z scores (critical values) that corresponds to a level of confidence of 98.22.
10. Use the given degree of confidence and sample data to construct a confidence interval
for the population mean. A random sample of 86 light bulbs had a mean life of 545
hours with a standard deviation of 29 hours. Construct a 90 percent confidence
interval for the mean life, , of all light bulbs of this type.
11. 40 packages are randomly selected from packages received by a parcel service. The
sample has a mean weight of 24.1 pounds and a standard deviation of 2.9 pounds.
What is the 95 percent confidence interval for the true mean weight, , of all
packages received by the parcel service?
12. 30 students were randomly selected from a large group of students taking a certain
calculus test. The mean score for the students in the sample was 81 with a standard
deviation of 6.9. Construct a 99% confidence interval for the mean score, , of all
students taking the test.
13. Find the critical values for a t-distribution for a confidence level of 99% and a sample
size of 17.
14. Find the critical values for the t-distribution that corresponds to a confidence level of
90% and a sample size of 10.
15. Find the critical values for the t-distribution that corresponds to a confidence level of
95% and a sample size of 11.
16. For the binomial distribution with the given values for n and p, state whether or not it
is suitable to use the normal distribution as an approximation.
a) n = 22 and p = .6
b) n = 20 and p = .8
17. A multiple choice test consists of 60 questions. Each question has 4 possible answers
of which one is correct. If all answers are random guesses, estimate the probability of
getting at least 20% correct.
18. A certain question on a test is answered correctly by 22% of the respondents.
Estimate the probability that among the next 150 responses there will be at most 40
correct answers.
19. The probability that a radish seed will germinate is 0.7. Estimate the probability that
of 140 randomly selected seeds, exactly 100 will germinate.
20. Find the probability that in 200 tosses of a fair die, we will obtain at least 40 fives.
21. Define a point estimate. What is the best point estimate for ?
22. Under what circumstances can you replace with s in the formula for E.
23. How do you determine whether to use the z or t distribution in computing the margin
of error?
24. What assumption about the parent population is needed to use the t distribution to
compute the margin of error?
25. Under what three conditions is it appropriate to use the t distribution in place of the
standard normal distribution?
26. A certain grade egg must weigh at least 2 oz. If the weights of eggs are normally
distributed with a mean of 1.5 oz and a standard deviation of 0.4 oz, approximately
how many eggs in a random sample of 22 dozen would you expect to weigh more
than 2 oz?
27. The diameters of pencils produced by a certain machine are normally distributed with
a mean of 0.30 inches and a standard deviation of 0.01 inches. In a random sample of
34 pencils, approximately how many would you expect to have a diameter less than
0.293 inches?
28. Scores on a test are normally distributed with a mean of 66.2 and a standard deviation
of 9. Find the value of P81.
29. The amount of rainfall in January in a certain city is normally distributed with a mean
of 4.7 and a standard deviation of 5.1 inches. Find the value of the quartile Q1.
30. Scores on an English test are normally distributed with a mean of 33.7 and a standard
deviation of 7.2. Find the score that separates the top 59% from the bottom 41%.
31. Suppose that replacement times for washing machines are normally distributed with a
mean of 10.6 years and a standard deviation of 2 years. Find the replacement time that
separates the top 18% from the bottom 82%.
32. The weights of the fish in a certain lake are normally distributed with a mean of 14 lb
and a standard deviation of 6 lb. If 4 fish are randomly selected, what is the
probability that the mean weight will be between 11.6 and 17.6 lb?
33. Assume that women's heights are normally distributed with a mean of 63.6 inches and
a standard deviation of 2.5 inches. If 90 women are randomly selected, find the
probability that they have a mean height between 62.9 inches and 64.0 inches.
34. A study of the amount of time it takes a mechanic to rebuild the transmission for a
1992 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation
is 1.8 hours. If 40 mechanics are randomly selected, find the probability that their
mean rebuild time exceeds 8.7 hours.
35. If Z is a standard normal variable, find the probability.
a) P(-0.73 < Z < 2.27)
b)P(Z < 0.97)
36. The Precision Scientific Instrument Company manufactures thermometers that are
supposed to give readings of 0C at the freezing point of water. Tests on a large
sample of these thermometers reveal that at the freezing point of water, some give
readings below 0C (denoted by negative numbers) and some give readings above
0C (denoted by positive numbers). Assume that the mean reading is 0C and the
standard deviation of the readings is 1. 0C. Also assume that the frequency
distribution of errors closely resembles the normal distribution. A thermometer is
randomly selected and tested. Find the temperature reading corresponding to the
given information.
a) Find Q3, the third quartile.
b) If 6.3% of the thermometers are rejected because they have readings that are
too high and another 6.3% are rejected because they have readings that are too
low, find the two readings that are cutoff values separating the rejected
thermometers from the others.
37. Assume that X has a normal distribution, and find the indicated probability.
The mean is = 15.2 and the standard deviation is = 0.9.
Find P(14.3 < X < 16.1).
38. The diameters of bolts produced by a certain machine are normally distributed with a
mean of 0.30 inches and a standard deviation of 0.01 inches. What percentage of bolts
will have a diameter greater than 0.32 inches?
39. A bank's loan officer rates applicants for credit. The ratings are normally distributed
with a mean of 200 and a standard deviation of 50. If an applicant is randomly
selected, find the probability of a rating that is between 170 and 220.
40. Assume that the weights of quarters are normally distributed with a mean of 5.67 g
and a standard deviation 0.070 g. A vending machine will only accept coins weighing
between 5.48 g and 5.82 g. What percentage of legal quarters will be rejected?
41. Replacement times for T.V. sets are normally distributed with a mean of 8.2 years and
a standard deviation of 1.1 years (based on data from "Getting Things Fixed,"
Consumers Reports).
a) Find the probability that a randomly selected T.V. will have a replacement time
between 6.5 and 9.5 years.
b) Find the probability that a randomly selected T.V. will have a replacement time
between 9.5 and 10.5 years.
c) These two problems are solved almost exactly the same. Draw the diagram for
each and discuss the part of the solution that would be different in finding the
requested probabilities.
42. State the Central Limit theorem.
43. Under what conditions can we apply the results of the central limit theorem?
44. The typical computer random-number generator yields numbers in a uniform
distribution between 0 and 1 with a mean of 0.500 and a standard deviation of 0.289.
a) Suppose a sample of size 50 is randomly generated. Find the probability that
the mean is below 0.300.
b) Suppose a sample size of 15 is randomly generated. Find the probability that
the mean is below 0.300.
c) These two problems appear to be very similar. Only one can be solved by the
Central Limit theorem. Which one and why?
45. Complete the following table for a distribution in which = 16. It might be helpful to
make a diagram to help you determine the continuity factor for each entry.
x is at least 12
x is at most 12
x is more than 12
x is less than 12
46. According to data from the American Medical Association, 10% of us are left-
handed. Suppose groups of 500 people are randomly selected. Find the probability
that at least 80 are left-handed. Describe the characteristics of this problem which
help you to recognize that the problem is about a binomial distribution which you are
to solve by estimating with the normal distribution. (Assume that you would not use a
computer, a table, or the binomial probability formula.)
47. If Z is a standard normal variable, find the probability.
a) The probability that Z lies between 0.7 and 1.98
b) The probability that Z lies between -0.55 and 0.55
c) The probability that Z is greater than -1.82
48. Suppose you are asked to find the 20th percentile and the 80th percentile for a set of
scores. These two problems are solved almost exactly the same. Draw the diagram for
each and discuss the part of the solution that would be different in finding the
requested probabilities.