Introduction to Statistics
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Problem 1
Although there is some controversy regarding the appropriateness of IQ scores as a measure of intelligence, IQ scores are commonly used for a variety of purposes. One commonly used IQ scale has a mean of 100 and a standard deviation of 15, and scores are approximately normally distributed. If we define
x = IQ score of a randomly selected individual
a. What is the probability that a random person will have a score greater than 90? Between 90 and 110?
b. One way to become eligible for membership in Mensa, an organization purport- edly for those of high intelligence, is to have a Stanford- Binet IQ score above 130. What proportion of the population would qualify for Mensa membership?
Problem 2
The probability that a fluorescent bulb burns for at least 500 hours is 0.90. Of 8 such bulbs, find the probability that
a. All 8 burn for at least 500 hours.
b. At least one burns for at least 500 hours.
c. What is the expected value of the number of bulbs that burn for at least 500 hours? What is the variance of the number of bulbs that burn for at least 500 hours?
d. A store has 1000 fluorescent bulbs. Approximate the probability that at least 890 bulbs last 500 hours (use the continuity correction).
e. A store imported 60% of its fluorescent bulbs from China and 40% from Korea. It is known that 90% of the bulbs from China burns for at least 500 hours whereas it is 92% for the bulbs from Korea. A customer who bought a bulb from the store found that it last more than 500 hours. What is the probability that the bulb was imported from China?
Problem 3
An instructor knows from past experience that student examination scores have mean 77 and standard deviation 15. At present, the instructor is teaching two separate classes—one of size 81 and the other of size 121.
a. Approximate the probability that the average test score in the class of size 81 lies between 72 and 82.
b. Repeat (a) for the class of size 81.
c. Let X 81 and Y 121 be the average test score in the class of 81 and 121, respectively. What is the expected value and variance of
X 81 − Y 121 ? What is the distribution of X 81 − Y 121 ?
d. What is the approximate probability that the average test score in the class of size 81is higher than that in the class of size 121?
Problem 4
In the “3 Spot” version of the former California Keno lottery game, the player picked three numbers from 1 to 40. Ten possible winning numbers were then randomly selected. It cost $1 to play. The table shows the possible outcomes.
|
Number.of.Matches |
Amount.Won |
Probability |
|
3 |
$20 |
0.012 |
|
2 |
$2 |
1.370 |
|
0 or 1 |
$0 |
0.851 |
a. Compute the expected value of the amount won for this game. Interpret what it means.
b. Calculate the variance of this game.
c. A customer buys 5 tickets. Calculate the expected value and variance of the net gain
Problem 5
A television manufacturer claims that 90% of its TV sets will need no service during the first 3 years of operation. A consumer agency wishes to check this claim, so it obtains a random sample of n= 100 purchasers and found that 86 did not have the TV set repaired during the first 3 years after purchase.
a. Test the manufacturer claim at the 0.05 level.
b. What is the p-value for the test?
Problem 6
All the students at a certain country are to be given a psychological task. To determine the average time it will take a student to perform this task, a random sample of 36 students was chosen and each was given the task. If it took these students an average of 12.4 minutes to complete the task with a sample standard deviation of 3.0 minutes.
a. Find a 95% confidence interval estimate for the average time it will take all students in the school to perform this task.
b. Explain what happens to the width of the confidence interval when
◦ the sample size increases
◦ the confidence level is reduced to 90%
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