Calculate the sample variance and sample standard deviation for the following frequency distribution of heights in centimeters for a sample
Question
Calculate the sample variance and sample standard deviation for the following frequency distribution of heights in centimeters for a sample of 8-year-old boys. If necessary, round to one more decimal place than the largest number of decimal places given in the data.
Class Frequency120.6 – 123.6 17123.7 – 126.7 49126.8 – 129.8 29129.9 – 132.9 41133.0 – 136.0 35
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2022-01-07T14:12:07+00:00
2022-01-07T14:12:07+00:00 1 Answer
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Answer:
[tex]\bar X = \frac{\sum_{i=1}^n X_i f_i}{n}= \frac{22026.1}{171}=128.81[/tex]
[tex] s^2 = \frac{2839948.85- \frac{(22026.1)^2}{171}}{171-1}= 16.59[/tex]
[tex] s= \sqrt{16.587}=4.07[/tex]
Step-by-step explanation:
For this case we can calculate the sample variance and deviation with the following table
Class Midpoint (Xi) fi Xi*fi Xi^2 *fi
120.6-123.6 122.1 17 2075.7 253443
123.7-126.7 125.2 49 6134.8 768077
126.8-129.8 128.3 29 3720.7 477365.8
129.9-132.9 131.4 41 5387.4 707904.4
133.0-136.0 134.5 35 4007.5 633158.8
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Total 171 22026.1 2839948.85
For this case we can calculate the mean or expected value with the following formula:
[tex]\bar X = \frac{\sum_{i=1}^n X_i f_i}{n}= \frac{22026.1}{171}=128.81[/tex]
Now we can calculate the sample variance with the following formula:
[tex]s^2 =\frac{\sum f_i X^2_i -[\frac{(\sum X_i f_i)}{n}]^2}{n-1}[/tex]
And if we replace we got:
[tex] s^2 = \frac{2839948.85- \frac{(22026.1)^2}{171}}{171-1}= 16.59[/tex]
And the standard deviation would be the square root of the variance and we got:
[tex] s= \sqrt{16.59}=4.07[/tex]