6.6.1: Practice Problems Corequisite S.6

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The U.S. Census Bureau collects information on the racial composition of people in counties throughout the U.S. by asking survey respondents to identify their race. This information allows researchers to estimate the percentage of people in a county who are in various racial categories. The histogram below displays the percentage of adults who identified their race as white in 1,000 counties in the U.S. in 2020.²¹ The number above each bar is the frequency of the group of values. The midpoint of each bar is the position on the horizontal axis at the center of the bar.

$histogram showing 100 counties, frequency of white on y-axis and white on x-axis. y-axis has scale 0 to 500 in increments of 100. x-axis has scale 0 to 100 in increments of 20. 0-10 is 1. 10-20 is 10. 20-30 is 17. 30-40 is 28. 40-50 is 53. 50-60 is 64. 60-70 is 112. 70-80 is 106. 80-90 is 199. 90-100 is 410.$

(1) What is the shape of the distribution: skewed to the right, skewed to the left, or bell-shaped?

(2) Use the histogram to estimate the mean. Note: The mean can be estimated by a weighted mean. Write your answer as a percentage rounded to two decimal places.

(3) Use the histogram to estimate the median. Write your answer as a percentage.

(4) Use the histogram to estimate the mode. Write your answer as a percentage.

(5) Which measure (mean, median or mode) best represents the typical percentage of adults who are white in this sample of counties? Explain.

(6) What percentage of the counties in this sample have 80% or more white residents? Write your answer rounded to one decimal place.

(7) What percentage of the counties in this sample have a majority of non-white residents? Write your answer rounded to one decimal place.

The GPAs of 1,000 students at a university have a bell-shaped distribution with a mean of 3.1 and standard deviation 0.25. The GPAs are summarized in the histogram below. We can use the histogram to show how the distribution is approximately symmetric to the mean.

$Histogram showing university students’ GPAs. frequency of GPA on y-axis and GPA on x-axis. Y-axis scaled o to 160 in increments of 20. X-axis scaled 2.2 to 4.0 in increments of 0.2. 2.3 is 1. 2.4 is 2. 2.5 is about 17. 2.6 is about 18. 2.7 is about 43. 2.8 is about 80. 2.9 is about 120. 3.0 is about 145. 3.1 is about 160. 3.2 is about 158. 3.3 is about 118. 3.4 is about 70. 3.5 is about 50. 3.6 is about 18. 3.7 is about 8. 3.9 is about 8.$

(8) Find the approximate percentage of data values that are 0.3 or less from the mean. Write your answer as a percentage rounded to the nearest whole percent. Hint: The percentage of data values that are 0.3 or less from the mean is the percentage of values that are 2.8 or less.

(9) Find the approximate percentage of data values that are 0.3 or more from the mean. Write your answer as a percentage rounded to the nearest whole percent. Hint: The percentage of data values that are 0.3 or less from the mean is the percentage of values that are 3.4 or more.

Find and interpret the Z-scores of the GPAs of the following three students.

(10) Jayden: GPA = 3.4

(11) Lisa: GPA = 3.7

(12) Mark: GPA = 2.6

(13) Which of the three GPAs can be classified as unusual?

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²¹ https://mtgis-portal.geo.census.gov/arcgis/apps/MapSeries/index.html?appid=2566121a73de463995ed2b2fd7ff6eb7

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