Question

calculating variance and population standard deviation the average estimated hours a person in the united states spent playing video games per year from 2002 to 2012 were 71, 80, 82, 78, 80, 91, 107, 121, 125, 131, and 142. use the statistics calculator to find the variance and population standard deviation. round answers to the nearest whole number. the variance of this data set is blank. the population standard deviation for this data set is blank.

Explanation:

Step1: Calculate the mean ($\mu$)

First, we find the sum of the data set. The data points are 71, 80, 82, 78, 80, 91, 107, 121, 125, 131, 142.
The number of data points ($N$) is 11.
Sum ($\sum x$) = $71 + 80 + 82 + 78 + 80 + 91 + 107 + 121 + 125 + 131 + 142$
= $71+80 = 151$; $151+82 = 233$; $233+78 = 311$; $311+80 = 391$; $391+91 = 482$; $482+107 = 589$; $589+121 = 710$; $710+125 = 835$; $835+131 = 966$; $966+142 = 1108$? Wait, no, let's recalculate:

Wait, 71 + 80 = 151; 151 + 82 = 233; 233 + 78 = 311; 311 + 80 = 391; 391 + 91 = 482; 482 + 107 = 589; 589 + 121 = 710; 710 + 125 = 835; 835 + 131 = 966; 966 + 142 = 1108? Wait, that can't be right. Wait, 71, 80, 82, 78, 80, 91, 107, 121, 125, 131, 142. Let's count the numbers: 71 (1), 80 (2), 82 (3), 78 (4), 80 (5), 91 (6), 107 (7), 121 (8), 125 (9), 131 (10), 142 (11). So 11 numbers.

Let's add again:

71 + 80 = 151

151 + 82 = 233

233 + 78 = 311

311 + 80 = 391

391 + 91 = 482

482 + 107 = 589

589 + 121 = 710

710 + 125 = 835

835 + 131 = 966

966 + 142 = 1108? Wait, 966 + 142: 966 + 100 = 1066, 1066 + 42 = 1108. So sum is 1108.

Mean $\mu = \frac{\sum x}{N} = \frac{1108}{11} \approx 100.727$

Step2: Calculate the squared differences from the mean

For each data point $x_i$, calculate $(x_i - \mu)^2$:

• For 71: $(71 - 100.727)^2 = (-29.727)^2 \approx 883.69$
• For 80: $(80 - 100.727)^2 = (-20.727)^2 \approx 429.61$
• For 82: $(82 - 100.727)^2 = (-18.727)^2 \approx 350.70$
• For 78: $(78 - 100.727)^2 = (-22.727)^2 \approx 516.52$
• For 80: $(80 - 100.727)^2 = (-20.727)^2 \approx 429.61$
• For 91: $(91 - 100.727)^2 = (-9.727)^2 \approx 94.62$
• For 107: $(107 - 100.727)^2 = (6.273)^2 \approx 39.35$
• For 121: $(121 - 100.727)^2 = (20.273)^2 \approx 410.99$
• For 125: $(125 - 100.727)^2 = (24.273)^2 \approx 589.18$
• For 131: $(131 - 100.727)^2 = (30.273)^2 \approx 916.46$
• For 142: $(142 - 100.727)^2 = (41.273)^2 \approx 1703.46$

Step3: Calculate the sum of squared differences ($\sum (x_i - \mu)^2$)

Now, sum all these squared differences:

883.69 + 429.61 = 1313.3

1313.3 + 350.70 = 1664

1664 + 516.52 = 2180.52

2180.52 + 429.61 = 2610.13

2610.13 + 94.62 = 2704.75

2704.75 + 39.35 = 2744.1

2744.1 + 410.99 = 3155.09

3155.09 + 589.18 = 3744.27

3744.27 + 916.46 = 4660.73

4660.73 + 1703.46 = 6364.19

Step4: Calculate the variance ($\sigma^2$)

Variance for a population is $\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$

So $\sigma^2 = \frac{6364.19}{11} \approx 578.56$

Rounding to the nearest whole number, variance is 579.

Step5: Calculate the population standard deviation ($\sigma$)

Standard deviation is the square root of variance: $\sigma = \sqrt{\sigma^2}$

$\sigma = \sqrt{578.56} \approx 24.05$

Rounding to the nearest whole number, standard deviation is 24.

Answer:

The variance of this data set is $\boxed{579}$.

The population standard deviation for this data set is $\boxed{24}$.