In 2013, several new cars were randomly selected that had city gas mileages of 4040​ to 5050​ miles per gallon. The price of each selected car is recorded in the table below.




Car Prices ($)($)

35,92335,923

38,61638,616

29,90229,902

33,39633,396

26,81926,819

32,70432,704

28,77828,778

25,42225,422

32,06032,060

36,14236,142

38,38238,382

39,70039,700

32,66932,669

33,52933,529

26,49826,498

​

a) Find the population variance​ of the data set. Round to one decimal place.

Variance

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b) Find the population standard deviation​ of the data set. Round to one decimal place.

To calculate the **population variance** and **population standard deviation** for the given car prices, we follow these steps:

### Step 1: List of Car Prices
The car prices provided are:

\[
35,923, \ 38,616, \ 29,902, \ 33,396, \ 26,819, \ 32,704, \ 28,778, \ 25,422, \ 32,060, \ 36,142, \ 38,382, \ 39,700, \ 32,669, \ 33,529, \ 26,498
\]

### Step 2: Calculate the Population Mean
To find the mean $ \mu $, sum all the car prices and divide by the total number of cars (which is 15).

\[
\mu = \frac{\sum \text{Car Prices}}{n}
\]

### Step 3: Calculate the Squared Differences
For each price, subtract the mean $ \mu $, square the result, and sum these squared differences.

### Step 4: Calculate the Population Variance
The population variance $ \sigma^2 $ is given by:

\[
\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}
\]

### Step 5: Calculate the Population Standard Deviation
The population standard deviation $ \sigma $ is the square root of the variance:

\[
\sigma = \sqrt{\sigma^2}
\]

I'll now calculate the values using the provided data.### a) Population Variance:
The population variance of the car prices is approximately $ 19,288,835.8 $.

### b) Population Standard Deviation:
The population standard deviation of the car prices is approximately $ 4,391.9 $.

Would you like further details on the steps or calculations?

Here are five related questions to expand your understanding:

1. How is the population variance different from sample variance?
2. Why is squaring the differences in variance calculation important?
3. How do we interpret the standard deviation in terms of car prices?
4. What happens if we calculate the sample variance instead of the population variance?
5. Can variance or standard deviation be negative?

**Tip**: Standard deviation gives a direct measure of how spread out the values are from the mean, so it's useful for comparing consistency across datasets.

In 2013, several new cars were randomly selected that had city gas mileages of 40 to 50 miles per gallon. The price of each selected car is recorded in the table below:

Car Prices ($):
35,923, 38,616, 29,902, 33,396, 26,819, 32,704, 28,778, 25,422, 32,060, 36,142, 38,382, 39,700, 32,669, 33,529, 26,498.

(a) Find the population variance of the data set. Round to one decimal place.
(b) Find the population standard deviation of the data set. Round to one decimal place.

Calculate the population variance and standard deviation for a dataset of car prices ranging from $25,422 to $39,700. Learn key statistical concepts like variance, standard deviation, and population mean. Step-by-step solutions included, suitable for high school students in Grades 10-12.

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