Download 1. What requirements are necessary for a normal probability

1.
What requirements are necessary for a normal probability distribution to be a standard
normal probability distribution? The mean must have a mean of 0 and a standard
deviation of 1.
2.
What is the difference between a standard normal distribution and a nonstandard normal
distribution? The standard normal distribution is the normal distribution with a mean
of zero and a variance of one.
3.
Find the indicated values for the following problems:
a. z0.05 1.65
b. z0.01 2.33
c. z0.10 1.28
d. z0.02 2.05
4.
Based on data from the National Health Survey, men’s heights are normally distributed
with mean 69.0 in. and standard deviation 2.8 in, whereas women’s heights are normally
distributed with mean 63.6 in. and standard deviation 2.5 in. The Gulfstream 100 is an
executive jet that seats six, and it has a doorway height of 51.6 in. Use this information to
answer the following problems:
a. What percentage of adult men can fit through the door without bending? The height of
the door corresponds to a z-score of (51.6 - 69.0)/(2.8) = -6.2, which means
approximately 0% of men can go through without bending.
b. What percentage of adult women can fit through the door without bending? The
height of the door for women corresponds to a z-score of (51.6 - 63.6)/(2.5) = -4.8,
which means approximately 0% of women can go through without bending.
c. Does the door design with a height of 51.6 in. appear to be adequate? Why didn’t the
engineers design a larger door? No, since pretty much nobody is 4' 3.6" tall or
shorter. Perhaps they only had limited space on which to fit the door.
d. What doorway height would allow 60% of men to fit without bending? If 60% of men
are to fit without bending, this means a z-score of 0.253, so the door needs to have
a height of 63.6 + (0.253)(2.8) = 64.3 inches.
5.
What is the standard error of the mean? The standard error is the estimated standard
deviation or measure of variability in the sampling distribution of a statistic. A low
standard error means there is relatively less spread in the sampling distribution. The
standard error indicates the likely accuracy of the sample mean as compared with the
population mean. The standard error decreases as the sample size increases and
approaches the size of the population. Sigma (σ) denotes the standard error; a
subscript indicates the statistic.
6.
Assume that SAT scores are normally distributed with mean μ = 1518 and standard
deviation σ = 325. Use the Central Limit Theorem to answer the following:
a. If 1 SAT score is randomly selected, find the probability that it is greater than 1600.
P(x > 1600) = P((x-μ)/σ > (1600-1518)/325) = P(z > .25) = .4013
b. If 64 SAT scores are randomly selected, find the probability that they have a mean
greater than 1600. 0.0218
7.
Assume that a sample is used to estimate a population proportion p. Find the margin of
error E that corresponds to the given statistics and confidence level.
a. n = 500, x = 220, 99% confidence
b. 90% confidence; the sample size is 1780, of which 35% are successes
8.
Use the given sample data and confidence level to construct the confidence interval
estimate of the population proportion p for each problem.
a. n = 2000, x = 400, 95% confidence
b. n = 5200, x = 4821, 99% confidence
9.
A design engineer for the Ford Motor Company must estimate the mean leg length of all
adults. She obtains a list of the 1275 employees at her facility, then obtains a simple
random sample of 50 employees. If she uses this sample to construct a 95% confidence
interval to estimate the mean leg length for the population of all adults, will her estimate be
good? Why or why not? No. The list of the employees at her facility from which she
obtained her simple random sample is itself a convenience sample. Those employees
are likely not representative of the population age, gender, ethnicity, or other factors
that may affect leg length
10.
A simple random sample of 40 salaries of NCAA football coaches has a mean of $415,953
and a standard deviation of $463,364.
a. Find the best point estimate of the mean salary of all NCAA football coaches. Point
estimate is also $415,953
b. Construct a 95% confidence interval estimate of the mean salary of an NCAA football
coach. 95% CI is 415,953 +/- 1.96(463,364)/sqrt(40), or 415,953 +/- 143,598,
or from $272,355 to $559,551
c. Does the confidence interval contain the actual population mean of $474,477? Yes. In
this case the confidence interval includes the true population mean.
11.
Use the given confidence level and sample data below to find (a) the margin of error and
(b) the confidence interval for the population mean. Assume that the sample is a simple
random sample and the population has a normal distribution. 99% confidence; n = 7, 𝑥 =
0.12 , s = 0.04
12.
What is an unbiased estimator? Is the sample variance an unbiased estimator of the
population variance? Is the sample standard deviation an unbiased estimator of the
population standard deviation? In statistics, the bias of an estimator is the difference
between this estimator’s expected value and the true value of the parameter being
estimated. An estimator or decision rule with zero bias is called unbiased.
13.
Use the given confidence level and sample data to find a confidence interval for the
population standard deviation. In each case, assume that a simple random sample has been
selected from a population that has a normal distribution. 95% confidence; n = 25, 𝑥 =
81.0, s = 2.3
14.
Twelve different video games showing substance use were observed and the duration times
of game play (in seconds) are listed below (based on data from “Content and Ratings of
Teen-Rated Video Games,” by Haninger and Thompson, Journal of the American Medical
Association, Vol. 291, No. 7). The design of the study justifies the assumption that the
sample can be treated as a simple random sample. 4049 3884 3859 4027 4318 4813 4657
4033 5004 4823 4334 4317
a. Use the sample data to construct a 95% confidence interval estimate of μ, the mean
duration of game play.
b. Use the sample data to construct a 99% confidence interval estimate of σ, the standard
deviation of the duration times of game play.