Question

1. ... the weekly numbers of ... worked on the summer job? if the mean of the data is 20 and a standard deviation of 5.7 ... the data: {11, 14, 18, 20, 23, 24, 25, 27}

Explanation:

Step1: Find the mean ($\mu$)

First, sum all the data points: $11 + 14 + 18 + 20 + 23 + 24 + 25 + 27$.
Sum calculation: $11 + 14 = 25$; $25 + 18 = 43$; $43 + 20 = 63$; $63 + 23 = 86$; $86 + 24 = 110$; $110 + 25 = 135$; $135 + 27 = 162$.
There are $n = 8$ data points. So, $\mu=\frac{162}{8}=20.25$.

Step2: Calculate each deviation from the mean and square it

• For $11$: $(11 - 20.25)^2 = (-9.25)^2 = 85.5625$
• For $14$: $(14 - 20.25)^2 = (-6.25)^2 = 39.0625$
• For $18$: $(18 - 20.25)^2 = (-2.25)^2 = 5.0625$
• For $20$: $(20 - 20.25)^2 = (-0.25)^2 = 0.0625$
• For $23$: $(23 - 20.25)^2 = (2.75)^2 = 7.5625$
• For $24$: $(24 - 20.25)^2 = (3.75)^2 = 14.0625$
• For $25$: $(25 - 20.25)^2 = (4.75)^2 = 22.5625$
• For $27$: $(27 - 20.25)^2 = (6.75)^2 = 45.5625$

Step3: Find the sum of squared deviations

Sum these squared deviations: $85.5625 + 39.0625 + 5.0625 + 0.0625 + 7.5625 + 14.0625 + 22.5625 + 45.5625$.
Calculation: $85.5625+39.0625 = 124.625$; $124.625+5.0625 = 129.6875$; $129.6875+0.0625 = 129.75$; $129.75+7.5625 = 137.3125$; $137.3125+14.0625 = 151.375$; $151.375+22.5625 = 173.9375$; $173.9375+45.5625 = 219.5$.

Step4: Calculate variance ($\sigma^2$) and standard deviation ($\sigma$)

Variance for population is $\sigma^2=\frac{\sum (x_i - \mu)^2}{n}$. So, $\sigma^2=\frac{219.5}{8}=27.4375$.
Standard deviation is the square root of variance: $\sigma=\sqrt{27.4375}\approx5.24$.

Answer:

Mean: $20.25$, Variance: $27.4375$, Standard Deviation: $\approx5.24$