Question

starting teachers’ salaries starting teacher salaries for upper secondary education in selected states are listed below.
starting salaries average salaries
iowa 27,284 iowa 41,083
kansas 27,840 kansas 41,467
kentucky 30,619 kentucky 42,592
louisiana 31,298 louisiana 40,029
maine 26,643 maine 40,737
maryland 37,125 maryland 54,333
massachusetts 35,421 massachusetts 56,369
michigan 35,557 michigan 54,739
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part 1 of 3
find the range, variance, and standard deviation of starting salaries for 8 states. round your answers to at least one decimal place.
range = 10482
variance = 17006601.8
standard deviation = 4123.9
correct answer
range = 10482
variance = 17906587.7
standard deviation = 4123.9
part: 1 / 3
part 2 of 3
find the range, variance, and standard deviation of average salaries for 8 states. round your answers to at least one decimal place.
range =
variance =
standard deviation =

Explanation:

Step1: Find the range of average salaries

The range is calculated as the maximum value minus the minimum value. First, identify the maximum and minimum average salaries from the data:

• Average salaries: 41,083 (Iowa), 41,467 (Kansas), 42,592 (Kentucky), 40,029 (Louisiana), 40,737 (Maine), 54,333 (Maryland), 56,369 (Massachusetts), 54,739 (Michigan)
• Maximum value = 56,369
• Minimum value = 40,029
• Range = 56,369 - 40,029 = 16,340

Step2: Calculate the mean of average salaries

The mean ($\bar{x}$) is the sum of all values divided by the number of values (n = 8).

• Sum of average salaries: 41,083 + 41,467 + 42,592 + 40,029 + 40,737 + 54,333 + 56,369 + 54,739
• 41,083 + 41,467 = 82,550
• 82,550 + 42,592 = 125,142
• 125,142 + 40,029 = 165,171
• 165,171 + 40,737 = 205,908
• 205,908 + 54,333 = 260,241
• 260,241 + 56,369 = 316,610
• 316,610 + 54,739 = 371,349
• Mean = $\frac{371,349}{8}$ = 46,418.625

Step3: Calculate the variance of average salaries

Variance ($s^2$) is calculated as $\frac{\sum (x_i - \bar{x})^2}{n - 1}$.

• Calculate each $(x_i - \bar{x})^2$:
• For Iowa: $(41,083 - 46,418.625)^2 = (-5,335.625)^2 = 28,479,095.140625$
• For Kansas: $(41,467 - 46,418.625)^2 = (-4,951.625)^2 = 24,518,590.140625$
• For Kentucky: $(42,592 - 46,418.625)^2 = (-3,826.625)^2 = 14,643,062.140625$
• For Louisiana: $(40,029 - 46,418.625)^2 = (-6,389.625)^2 = 40,827,290.140625$
• For Maine: $(40,737 - 46,418.625)^2 = (-5,681.625)^2 = 32,280,862.140625$
• For Maryland: $(54,333 - 46,418.625)^2 = (7,914.375)^2 = 62,637,330.140625$
• For Massachusetts: $(56,369 - 46,418.625)^2 = (9,950.375)^2 = 99,009,950.140625$
• For Michigan: $(54,739 - 46,418.625)^2 = (8,320.375)^2 = 69,228,630.140625$
• Sum of $(x_i - \bar{x})^2$:
• 28,479,095.140625 + 24,518,590.140625 = 52,997,685.28125
• 52,997,685.28125 + 14,643,062.140625 = 67,640,747.421875
• 67,640,747.421875 + 40,827,290.140625 = 108,468,037.5625
• 108,468,037.5625 + 32,280,862.140625 = 140,748,899.703125
• 140,748,899.703125 + 62,637,330.140625 = 203,386,229.84375
• 203,386,229.84375 + 99,009,950.140625 = 302,396,179.984375
• 302,396,179.984375 + 69,228,630.140625 = 371,624,810.125
• Variance = $\frac{371,624,810.125}{8 - 1}$ = $\frac{371,624,810.125}{7}$ ≈ 53,089,258.6

Step4: Calculate the standard deviation of average salaries

The standard deviation (s) is the square root of the variance.

• Standard deviation = $\sqrt{53,089,258.6}$ ≈ 7,286.2

Answer:

• Range = 16,340.0
• Variance ≈ 53,089,258.6
• Standard deviation ≈ 7,286.2