Normal Distribution Examples

Normal Distribution Problems
1. z is normally distributed with mean 0 and standard deviation 1. What is p(−.5 ≤ z ≤ 1.0)?
2. x is normally distributed with mean 10 and standard deviation 2. What is p(11.0 ≤ x ≤ 13.6)?
3. Studies show that gasoline usage for compact cars sold in the United States is normally distributed
with a mean of 25.5 miles per gallon and a standard deviation of 6.5 miles per gallon.
What percentage of compacts obtain 30 or more miles per gallon?
4. Refer to previous question. If a manufacturer wishes to develop a compact car which outperforms
95% of the current compacts in fuel economy, what must be the gasoline usage rate for
the new car?
5. The length of a type of automatic washer is approximately normally distributed with mean
and standard deviation of 3.1 and 1.2 years, respectively. If this type of washer is guaranteed
for one year, what fraction of original sales will require replacement?
6. Same problem. What should the warranty period be changed to if the company is willing to
replace 15% of the washers?
7. A venture capital company feels that the rate of return on a proposed investment is approximately
normally distributed with a mean of 30% and a standard deviation of 10%. What is
the probability that the return will exceed 55%? What is the probability that the return will
be less than 22%?
8. A muffler company advertises that you will receive a rebate if it takes longer than 30 minutes
to replace your muffler. Experience has shown that the time taken to replace a muffler is
approximately normally distributed with a mean of 25 minutes and a standard deviation of 5
minutes. What proportion of customers receive a rebate?
9. Same problem. What proportion of mufflers take between 22 and 26 minutes to replace?
10. Same problem. What should the rebate–determining time of 30 minutes be changed to if the
company wishes to provide only 1% of its customers with a rebate?
11. The daily sales (excepting Saturday) at a small restaurant has a probability distribution that
is approximately normal with mean equal to $530 per day and standard deviation equal to
$120. What is the probability that the sales will exceed $700 on a given day? The restaurant
must have at least $300 per day in order to break even. What is the probability that on a
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given day the restaurant will not break even?
12. Suppose that the counts on the number of a particular type of bacteria in 1 milliliter of
drinking water tend to be approximately normally distributed with a mean of 85 and a standard
deviation of 9. What is the probability that a given 1 milliliter sample will contain more than
100 bacteria?
13. A professional painter has found that the area he can usually cover with a gallon of paint
is normally distributed with a mean of 400 sq. ft. and a standard deviation of 50 ft. A
manufacturer advertises that 1 gallon of his paint will cover between 375 sq. ft. and 450 sq.
ft. What is the probability that the area covered by this paint will be within the advertised
limits? What is the probability that the paint will cover more area than advertised?
14. z is normally distributed with mean 0 and standard deviation 1. Verify the Empirical Rule by
finding the following probabilities: p(−1 ≤ z ≤ 1), p(−2 ≤ z ≤ 2), and p(−3 ≤ z ≤ 3).
15. x is normally distributed. Pick any mean µ and standard deviation σ you wish. Verify the
Empirical Rule by finding the following probabilities:
• p(µ − σ ≤ x ≤ µ + σ)
• p(µ − 2σ ≤ x ≤ µ + 2σ)
• p(µ − 2σ ≤ x ≤ µ + 2σ)
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Fact 2 (Central Limit Theorem):
If x is any random variable with mean µ x and standard deviation σ x , then the sample mean x
for samples of size N, is approximately normally distributed with mean µ x = µ x and standard
deviation σ x = σ x
√
N
.
We will use this fact in two ways:
i) We will assume we know µ x and σ x and calculate various probabilities involving x.
ii) We will estimate an unknown population mean using x for a particular sample of size N.
We will attach levels of confidence to our estimate and construct confidence intervals for
the population mean of the form
x + − something = (x − something , x + something) .
For now we’ll concentrate on the first type of problem.
Problems:
1. For a certain population, µ x = 50 and σ 2 x = 64. A single sample of size 25 is drawn from this
population. What is the probability that the mean for this sample is 54 or less?
2. Same problem. What is the probability that the sample mean deviates by no more than 2
units in either direction from the population mean?
3. Same problem. The probability is 0.05 that the sample mean will be less than what value?
4. For a certain population of people (yeah, you guessed it: cat lovers), the average number of
cats in a household is 12 with a standard deviation of 9. A random survey of 16 of these foolish
households is performed. What is the probability that the mean for this sample is more than
10?
5. Same problem. What is the probability that the sample mean is between 13 and 15?
6. Same problem. The probability is 15% that the sample mean will be less than what value?
7. Same problem. What is the probability I hung up the telephone and played a game called Cat
Splat on my pc as soon as I heard the word cat when the people doing the survey called.
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