BackTesting Goodness-of-Fit with a Normal Distribution Refer to Data Set 1 “Body Data” in Appendix B for the heights of females.
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c. Using the probabilities found in part (b), find the expected frequency for each category.
Basis for the Range Rule of Thumb and the Empirical Rule. In Exercises 45–48, find the indicated area under the curve of the standard normal distribution; then convert it to a percentage and fill in the blank. The results form the basis for the range rule of thumb and the empirical rule introduced in Section 3-2.
About __ % of the area is between z = -1 and z = 1 (or within 1 standard deviation of the mean).
Manufacturer Claims You work for a consumer watchdog publication and are testing the advertising claims of a tire manufacturer. The manufacturer claims that the life spans of the tires are normally distributed, with a mean of 40,000 miles and a standard deviation of 4000 miles. You test 16 tires and record the life spans shown below.
a. Draw a frequency histogram to display these data. Use five classes. Do the life spans appear to be normally distributed? Explain.
app’s Index of Skewness The English statistician Karl app (1857–1936) introduced a formula for the skewness of a distribution.
P = 3 (x̄ - median) / s
Most distributions have an index of skewness between -3 and 3. When P > 0, the data are skewed right. When P < 0, the data are skewed left. When P = 0, the data are symmetric. Calculate the coefficient of skewness for each distribution. Describe the shape of each.
a. x̄ = 17, s = 2.3, median = 19
app’s Index of Skewness The English statistician Karl app (1857–1936) introduced a formula for the skewness of a distribution.
P = 3 (x̄ - median) / s
Most distributions have an index of skewness between -3 and 3. When P > 0, the data are skewed right. When P < 0, the data are skewed left. When P = 0, the data are symmetric. Calculate the coefficient of skewness for each distribution. Describe the shape of each.
c. x̄ = 9.2, s = 1.8, median = 9.2
app’s Index of Skewness The English statistician Karl app (1857–1936) introduced a formula for the skewness of a distribution.
P = 3 (x̄ - median) / s
Most distributions have an index of skewness between -3 and 3. When P > 0, the data are skewed right. When P < 0, the data are skewed left. When P = 0, the data are symmetric. Calculate the coefficient of skewness for each distribution. Describe the shape of each.
e. x̄ = 155, s = 20.0, median = 175
Graphical Analysis In Exercises 19–22, use the box-and-whisker plot to determine whether the shape of the distribution represented is symmetric, skewed left, skewed right, or none of these. Justify your answer.
Graphical Analysis In Exercises 19–22, use the box-and-whisker plot to determine whether the shape of the distribution represented is symmetric, skewed left, skewed right, or none of these. Justify your answer.
Finding Area In Exercises 23–36, find the indicated area under the standard normal curve. If convenient, use technology to find the area.
To the left of z= -1.28 and to the right of z= 1.28
Given the mean of a normal distribution, how can you find the median?
Describe the inflection points on the graph of a normal distribution. At what x-values are the inflection points located?
The data set represents the number of minutes a sample of 27 people exercise each week.
108 139 120 123 120 132 123 131 131
157 150 124 111 101 135 119 116 117
127 128 139 119 118 114 127 142 130
d. Describe the shape of the distribution as symmetric, uniform, skewed left, skewed right, or none of these.
In Exercises 21 and 22, determine whether the approximate shape of the distribution in the histogram is symmetric, uniform, skewed left, skewed right, or none of these.
Describe the shape of the distribution for the histogram you made in Exercise 3 as symmetric, uniform, skewed left, skewed right, or none of these.
Draw two normal curves that have the same mean but different standard deviations. Describe the similarities and differences.