Exercises

Exercises for Section  4.1

1.

A study was conducted in order to determine if any link existed between cellular phone usage and the development of brain cancer. (Don't worry, no connection was found.) Data from this study indicate that the monthly phone usage for all users is approximately normally distributed with mean 2.4 hours and standard deviation 1.1 hours.

(a)

What proportion of cell phone users are on their phones between 1 hour and 3 hours per month?

(b)

Just to be safe, suppose you decide to be in the 5^th percentile of cell phone users in terms of monthly usage. How much time can you spend on your phone this month?

2.

On the average, a watch battery is known to last for 2 years (24 months), with a standard deviation of 9 months. Assuming that the population is normally distributed,

(a)

What percentage of watch batteries last more than 6 months?

(b)

What is the life span of a watch battery which lasts longer than 60% of all batteries?

(c)

What proportion of watch batteries last shorter than 2 years or longer than $3 \frac{1}{2}$ years (42 months)?

3.

Data from previous years reveal that the distribution of first exam scores in an introductory statistics class is approximately normal with a mean of 72 and a standard deviation of 12.

(a)

Given that the passing score is 65, approximately what percentage of students pass the first exam?

(b)

If a student aims to be at least in the 98^th percentile, how should he or she score?

(c)

The middle 95% of the scores will tend to fall between _________ and _________.

4.

Mastercard records show that credit card purchases for the month of December average $1200 with standard deviation $550. Assuming that purchases approximately follow the normal curve

(a)

What percentage of customers spend between $1000 and $2000 in December using their Mastercard?

(b)

What percentage of customers will charge less than $2000 for the month of December?

(c)

Suppose that Mastercard gives special offers to customers who spend more than $1500 in a month. What percentage of customers will ``earn'' a special offer for their December purchases?

(d)

How much did a Mastercard customer spend if only 2% of all customers spent more?

5.

Suppose 95% of data coming from a normally distributed population falls between 4 and 35. Based on the empirical rule, what is the standard deviation of this sample of data?

6.

Employees of a company are given a test that measures compentency. Scores from this test are known to be distributed normally with mean 200 and standard deviation 35. The people scoring in the bottom 3% will be fired. What score is necessary to retain a job?

7.

Let X be a normal random variable with $\mu = 25$ and $\sigma = 10$. Find the following probabilities.

(a)

P(X > 22)

(b)

P(24 < X < 39)

(c)

P(X < 18)

Exercises for Section  4.2

8.

Records in a Beanie Babies factory show that for every 100 newly produced toys, 20 are defective. If the quality control manager takes a sample of 10 newly manufactured stuffed toys,

(a)

What is the chance that she will find 5 or more defectives?

(b)

What is the likelihood that all stuffed toys are in perfect condition (no defectives)?

(c)

How many non-defectives should she expect to find?

(d)

Suppose 200 toys were sampled instead of 10. The number of defectives will be between what two values with 68% chance?

9.

NFL place kicker Jan Stenerud has a lifetime record of 66.2% for making field goals. Assume that his field goal attempts are independent. Out of five attempts, what is the probability he makes at least four of them?

10.

Suppose a student in Stat 216 has not been attending class this semester but decides to take the exam anyway. If he randomly guesses on each of the 25 questions, then he has a 1 out of 5 chance of getting a correct answer, since it is a multiple choice exam with choices a, b, c, d, or e.

(a)

How many questions should the student expect to get correct on this exam?

(b)

What is the probability that the student will get a "C" or better, which is 16 or more correct?

(c)

What is the probability that the student will score lower than a "C" (15 or less correct)?

11.

For each of the following situations, decide if the binomial distribution is appropriate. If yes, determine n and p. If not, explain why not.

(a)

A random sample of 10 newborn babies is taken from the nursery of the city hospital. Let X be the number of girls in the sample.

(b)

People getting off the Lovell street bus at the business college are asked if they own a dog. Let X be the number of people until the first dog owner gets off the bus.

(c)

Suppose the pond you're fishing in contains 25 bluegills and 10 bass. If you land 5 fish, let X be the number of bass among your catch.

12.

The quality control section of a purchasing contract for valves specifies that the diameter must be between 2.53 and 2.57 centimeters. Assume that the production equipment is set so that the mean diameter is 2.56 centimeters and the standard deviation is 0.01 cm.

(a)

What proportion of valves produced, over the long run, will be within these specifications, assuming a normal distribution?

(b)

What proportion of valves produced will fall outside the specifications?

13.

Millions of Americans are subjected to drug testing in the workplace. The tests are not always reliable. Foods and medicines such as Advil, Midol, Sudafed, poppy seeds, and some antibiotics can cause false-positive results. It has even been shown that the skin pigment melatonin causes those with dark skin to be at a higher risk of falsely testing positive. A false positive rate of 5% is considered to be a conservative estimate. Suppose that there are 40 employees in your company, which tests its workers weekly.

(a)

What is the probability of no false-positives for this week's test?

(b)

What is the probability of at least one false-positive for this week's test?

(c)

What is the likelihood that the test will give 5 to 10 false-positives?

(d)

What is the chance of at most 3 false-positives?

(e)

How many false-positives should the company expect?

Exercises for Section  4.3

14.

Suppose that it is a known fact that 75% of all college students are single. A random sample of 120 college students is taken for observation. Note that np = 120(.75) = 90 > 5, and n(1-p) = 120(.25) = 30 > 5, so the normal approximation to the binomial is valid.

(a)

What is the approximate probability that at least 85 students in the sample will be single?

(b)

What is the approximate probability that between 100 and 105 (inclusive) students in the sample will be single?

15.

It is known that 30% of all customers of a major credit card pay their bills in full before any interest charges are incurred. Suppose a simple random sample of 150 credit card holders is selected. Note that np = 150(.3) = 45 > 5, and n(1-p) = 150(.7) = 105 > 5, so the normal approximation to the binomial is quite valid.

(a)

What is the probability that 30 or fewer customers pay their account balances in full before any interest charges are incurred?

(b)

What is the probability that between 40 and 50 customers pay their account balances in full before any interest charges are incurred?

(c)

What is the probability that more than 60 customers pay their account balances in full before any interest charges are incurred?

16.

Based on past experience, 20% of all customers at a particular automotive service station pay for their purchases with cash or check. Supoose a random sample of 100 customers is selected for observation. Note that np = 100(.2) = 20 > 5, and n(1-p) = 100(.8) = 80 > 5, so the normal approximation to the binomial is valid.

(a)

What is the approximate probability that at least 32 customers pay with cash or check?

(b)

What is the approximate probability that between 10 and 25 customers pay with cash or check?

(c)

What is the approximate probability that 50 or more customers pay with cash or check?