# Normal Probability/ Testing Hypothesis

The listed sample distances (in millimeters) were obtained by using a pupilometer to measure the distances between the pupils of adults (based on data collected by a student of the author)

67 66 59 62 63 66 66 55

~

a) Find the mean x of the distances in this sample.

b) Find the median of the distances in this sample.

c) Find the mode of the distances in this sample.

d) Find the standard deviation s of this sample.

e) Convert the distance of 59mm to a z score.

f) Find the actual percentage of these sample values that exceed 59mm.

g) Assuming a normal distribution find the percentage of population distances that exceed 59 mm Use the sample values x and s as estimates of u and o

h) What level of measurement (nominal, ordinal, interval, ratio) describes this data set?

i) The listed measurements appear to be rounded to the nearest millimeter, but are the exact unrounded distances discrete data or continuous data?

According to data from the American Medical Association 10% of us are left-handed.

a) If three people are randomly selected find the probability that they are all left- handed.

b) If thee people are randomly selected find the probability that at least one of them is left-handed.

c) Why can't we solve the problem inpart (b) by using the normal approximation to the binomial distribution?

d) If groups of 50 people are randomly selected what is the mean number of left-handed people in such group.

e) If groups of 50 people are randomly selected what is the standard deviation for the number of left-handed people in such group

f) Would it be unusual to get 8 left-handed people in a randomly selected group of 50 people why or why not?

Listed below are measured amounts of dioxin in the air at the site of the World Trade Center on the days immediately following the terrorist attacks of September 11, 2001. Dioxin includes a group of chemicals produced from burning and some types of manufacturing. The listed amounts are in nanograms per cubix meter (ng/m^3) and they are in order with the earliest recorded values at the left. The data are provided by the US Environmental Protection Agency.

0.161 0.175 0.176 0.032 0.0524 0.044 0.018 0.0281 0.0268

a) Find the mean of this sample

b) Find the median

c) Find the standard deviation

d) Find the variance

e) Find the range

f) Construct a 95% confidence interval estimate of the population mean.

g) The EPA uses 0.16 ng/m^3 as its "screening level" which is "set to protect against significantly increased risks of cancer and other adverse health effects" Use a 0.05 significance level to test the claim that this sample comes from a population with a mean less than 0.16 ng/m^3

h) Is there any important characteristics of the data not addressed by the preceding results? If so what is it.

The math SAT scores for women are normally distributed with a mean of 496 and a standard deviation of 108.

a) if a woman who take the math portion of the SAT is randomly selected find the probability that her score is above 500

b) if five math SAT scores are randomly selected from the population of women who take the test find the probability that all five of the scores are above 500

c) if five women who take the math portion of the SAT are randomly selected find the probability that their mean is above 500

d) Find P the score separating the bottom 90% from the top 10%

90

#### Solution Summary

The solution addresses The listed sample distances (in millimeters) were obtained by using a pupilometer to measure the distances between the pupils of adults (based on data collected by a student of the author)

67 66 59 62 63 66 66 55

~

a) Find the mean x of the distances in this sample.

b) Find the median of the distances in this sample.

c) Find the mode of the distances in this sample.

d) Find the standard deviation s of this sample.

e) Convert the distance of 59mm to a z score.

f) Find the actual percentage of these sample values that exceed 59mm.

g) Assuming a normal distribution find the percentage of population distances that exceed 59 mm Use the sample values x and s as estimates of u and o

h) What level of measurement (nominal, ordinal, interval, ratio) describes this data set?

i) The listed measurements appear to be rounded to the nearest millimeter, but are the exact unrounded distances discrete data or continuous data?

According to data from the American Medical Association 10% of us are left-handed.

a) If three people are randomly selected find the probability that they are all left- handed.

b) If thee people are randomly selected find the probability that at least one of them is left-handed.

c) Why can't we solve the problem inpart (b) by using the normal approximation to the binomial distribution?

d) If groups of 50 people are randomly selected what is the mean number of left-handed people in such group.

e) If groups of 50 people are randomly selected what is the standard deviation for the number of left-handed people in such group

f) Would it be unusual to get 8 left-handed people in a randomly selected group of 50 people why or why not?

Listed below are measured amounts of dioxin in the air at the site of the World Trade Center on the days immediately following the terrorist attacks of September 11, 2001. Dioxin includes a group of chemicals produced from burning and some types of manufacturing. The listed amounts are in nanograms per cubix meter (ng/m^3) and they are in order with the earliest recorded values at the left. The data are provided by the US Environmental Protection Agency.

0.161 0.175 0.176 0.032 0.0524 0.044 0.018 0.0281 0.0268

a) Find the mean of this sample

b) Find the median

c) Find the standard deviation

d) Find the variance

e) Find the range

f) Construct a 95% confidence interval estimate of the population mean.

g) The EPA uses 0.16 ng/m^3 as its "screening level" which is "set to protect against significantly increased risks of cancer and other adverse health effects" Use a 0.05 significance level to test the claim that this sample comes from a population with a mean less than 0.16 ng/m^3

h) Is there any important characteristics of the data not addressed by the preceding results? If so what is it.

The math SAT scores for women are normally distributed with a mean of 496 and a standard deviation of 108.

a) if a woman who take the math portion of the SAT is randomly selected find the probability that her score is above 500

b) if five math SAT scores are randomly selected from the population of women who take the test find the probability that all five of the scores are above 500

c) if five women who take the math portion of the SAT are randomly selected find the probability that their mean is above 500

d) Find P the score separating the bottom 90% from the top 10%

90