Penn State University Statistics Normal Distribution Questions

Assignment 5More questions on normal distribution.
Answer all of the following questions. These involve the empirical rule, normal distribution with
formula, and inverse normal problems (See assignments 1-4). Submit for grade.
1.) Airplane Takeoff Time. The taxi and takeoff time for a commercial jet is normally distributed
with a mean of 8.5 minutes and a standard deviation of 2.5 minutes.
a.) Sketch this normal curve and label the mean and specific x values.
Use this distribution and the 68-95-99.7 (empirical) rule to answer the following questions.
b.) About what percent of jets have a taxi and takeoff time less than 8.5 minutes?
c.) About what percent of jets have taxi and takeoff times above 11 minutes?
d.) About what percent of jets have taxi and takeoff times between 3.5 and 13.5 minutes?
e.) About what percent of jets have taxi and takeoff times below 1 minute?
f.) About what percent of jets have taxi and takeoff times between 3.5 and 11 minutes?
2.) Use the standard normal proportions table to calculate the following probabilities based on
the standard normal curve. In each case, sketch a normal curve and shade in the
appropriate area.
a.) P( z  1.38) =
b.) P ( z  −1.25) =
c.) P(0  z  2.53) =
d.) P ( z  −0.16) =
e.) P(−0.44  z  0.44) =
3.) What formula do you use when calculating the number of standard deviations from the
mean of a random variable x for a normal distribution problem?
4.) The National Collegiate Athletic Association (NCAA) requires Division I athletes to score at
least 820 on combined mathematics and verbal parts of the SAT exam to compete in their
first college year. (Higher scores are required for students with poor high school grades.) In
2002, the scores of the 1.3 million students taking the SATs were approximately Normal
with mean 1020 and standard deviation 207.
a.) Sketch a normal curve and label the mean and specific x values.
b.) To determine what percent of all students that had a score less than 820, what formula
do you need?
c.) Substitute relevant values into the formula and solve for z.
d.) Use the probability notation involving the inequality in relation to z to find the
probability.
5.) Red Blood Cell Count Let x = red blood cell count in millions per cubic millimeter of whole
blood. For healthy females, x has an approximately normal distribution with mean  = 4.8
and standard deviation  = 0.3 .
a.) Sketch a normal curve and label the mean and specific x values.
b.) Determine the probability of a woman having a red blood cell count less than 4.35
million per cubic millimeter.
c.) What is the probability of a woman having a red blood cell count greater than 5.56
million per cubic millimeter?
6.) According to data from the National Health Survey the heights of adult women have a mean
of  = 63.6 inches and a standard deviation of  = 2.5 inches.
a.) Sketch a normal curve and label the mean and specific x values.
b.) The U.S. Army requires women’s heights to be between 58 and 80 inches. Find the
percentage of women meeting that height requirement. Are many women being denied
the opportunity to join the Army because they are too short or too tall? (Hint: For zvalues too large for the chart use an approximation of 1 for its probability).
c.) In order to fit into a Russian Soyuz spacecraft, an astronaut must have a height between
64.5 and 72 inches. What percentage of women meet that requirement?
7.) Budget Maintenance The amount of money spent weekly on cleaning, maintenance, and
repairs at a large restaurant was observed over a long period of time to be approximately
normally distributed, with a mean  = $615 and standard deviation  = $42 .
a.) If $646 is budgeted for next week, what is the probability that the actual costs will exceed
the budgeted amount?
b.) Inverse Normal Distribution How much should be budgeted for weekly repairs, cleaning,
and maintenance so that the probability that the budgeted amount will be exceeded in a
given week is only 0.10?