Question

question 2 of 6 the first four students to arrive for a first - period statistics class were asked how much sleep (to the nearest hour) they got last night. their responses were 7, 7, 8, and 10. calculate the standard deviation. $s_x=square$ hours (round to 2 decimal places.) interpret the standard deviation. in each interpretation, $s_x$ represents the value of the standard deviation. on average, the number of hours the students slept last night is $s_x$ hours. the number of hours that each student slept is $s_x$ hours away from the mean. the number of hours of sleep last night for the first four students to arrive in a first period statistics class typically varies from the mean by about $s_x$ hours. the range of the middle 50% of the sleep amounts for all students in the class is approximately $s_x$ hours.

Explanation:

Step1: Calculate the mean

The data set is \(7,7,8,10\). The mean \(\bar{x}=\frac{7 + 7+8 + 10}{4}=\frac{32}{4}=8\).

Step2: Calculate the squared - differences

For \(x_1 = 7\): \((7 - 8)^2=(-1)^2 = 1\)
For \(x_2 = 7\): \((7 - 8)^2=(-1)^2 = 1\)
For \(x_3 = 8\): \((8 - 8)^2=0^2 = 0\)
For \(x_4 = 10\): \((10 - 8)^2=2^2 = 4\)

Step3: Calculate the variance

The variance \(s^{2}=\frac{(7 - 8)^2+(7 - 8)^2+(8 - 8)^2+(10 - 8)^2}{4 - 1}=\frac{1+1 + 0+4}{3}=\frac{6}{3}=2\).

Step4: Calculate the standard deviation

The standard deviation \(s=\sqrt{s^{2}}=\sqrt{2}\approx1.41\).

Answer:

1.41
The number of hours of sleep last night for the first four students to arrive in a first - period statistics class typically varies from the mean by about \(s_x\) hours.