Math  /  Data & Statistics

QuestionWhat's your favorite TV show? The following table presents the numbers of viewers, in millions for the top 12 prime-time shows for the 2013-2014 and 2018-2019 seasons. The numbers of viewers include those who watched the program on any platform, including time-shifting up to seven days after the original telecast. \begin{tabular}{|l|ccccccccccc|c|} \hline Top TV Programs 2013-2014 & 12.1 & 12.6 & 14.0 & 20.0 & 21.5 & 13.0 & 13.6 & 12.2 & 19.8 & 15.2 & 16.0 & 12.7 \\ \hline Top TV Programs 20182019\mathbf{2 0 1 8 - 2 0 1 9} & 19.3 & 17.4 & 15.3 & 14.6 & 14.4 & 15.9 & 13.8 & 12.7 & 12.6 & 11.9 & 11.7 & 12.8 \\ \hline \end{tabular} Send data to Excel
Part: 0/30 / 3
Part 1 of 3 (a) Find the population standard deviation of the ratings for 2013-2014, Round the answer to two decimal places as needed.
The population standard deviation of the ratings for 2013-2014 is \square .

Studdy Solution

STEP 1

1. Population standard deviation is calculated using the formula for population data, not a sample.
2. The ratings provided are for the entire population of top 12 prime-time shows for the 2013-2014 season.
3. The formula for population standard deviation is σ=1Ni=1N(xiμ)2 \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2} , where N N is the number of data points, xi x_i are the individual data points, and μ \mu is the mean of the data points.
4. The result should be rounded to two decimal places.

STEP 2

1. Calculate the mean (μ \mu ) of the ratings for 2013-2014.
2. Calculate the squared deviations from the mean for each rating.
3. Sum the squared deviations.
4. Divide by the number of data points (N N ) to get the variance.
5. Take the square root of the variance to find the standard deviation.
6. Round the result to two decimal places.

STEP 3

Calculate the mean (μ \mu ) of the ratings for 2013-2014.
Given data: 12.1,12.6,14.0,20.0,21.5,13.0,13.6,12.2,19.8,15.2,16.0,12.7 12.1, 12.6, 14.0, 20.0, 21.5, 13.0, 13.6, 12.2, 19.8, 15.2, 16.0, 12.7
μ=112i=112xi=12.1+12.6+14.0+20.0+21.5+13.0+13.6+12.2+19.8+15.2+16.0+12.712 \mu = \frac{1}{12} \sum_{i=1}^{12} x_i = \frac{12.1 + 12.6 + 14.0 + 20.0 + 21.5 + 13.0 + 13.6 + 12.2 + 19.8 + 15.2 + 16.0 + 12.7}{12}

STEP 4

Perform the addition and divide by the number of data points (N=12 N = 12 ) to find the mean.
μ=182.71215.225 \mu = \frac{182.7}{12} \approx 15.225

STEP 5

Calculate the squared deviations from the mean for each rating.
(12.115.225)2,(12.615.225)2,(14.015.225)2,,(12.715.225)2 (12.1 - 15.225)^2, (12.6 - 15.225)^2, (14.0 - 15.225)^2, \ldots, (12.7 - 15.225)^2

STEP 6

Compute each squared deviation:
(12.115.225)2=9.773625 (12.1 - 15.225)^2 = 9.773625 (12.615.225)2=6.878025 (12.6 - 15.225)^2 = 6.878025 (14.015.225)2=1.500625 (14.0 - 15.225)^2 = 1.500625 (20.015.225)2=22.752025 (20.0 - 15.225)^2 = 22.752025 (21.515.225)2=39.452025 (21.5 - 15.225)^2 = 39.452025 (13.015.225)2=4.960625 (13.0 - 15.225)^2 = 4.960625 (13.615.225)2=2.640625 (13.6 - 15.225)^2 = 2.640625 (12.215.225)2=9.180625 (12.2 - 15.225)^2 = 9.180625 (19.815.225)2=21.005625 (19.8 - 15.225)^2 = 21.005625 (15.215.225)2=0.000625 (15.2 - 15.225)^2 = 0.000625 (16.015.225)2=0.600625 (16.0 - 15.225)^2 = 0.600625 (12.715.225)2=6.375625 (12.7 - 15.225)^2 = 6.375625

STEP 7

Sum the squared deviations to find the total sum of squares.
9.773625+6.878025+1.500625+22.752025+39.452025+4.960625+2.640625+9.180625+21.005625+0.000625+0.600625+6.375625=125.120000 9.773625 + 6.878025 + 1.500625 + 22.752025 + 39.452025 + 4.960625 + 2.640625 + 9.180625 + 21.005625 + 0.000625 + 0.600625 + 6.375625 = 125.120000

STEP 8

Divide the sum of squares by the number of data points (N=12 N = 12 ) to find the variance.
Variance=125.1200001210.426667 \text{Variance} = \frac{125.120000}{12} \approx 10.426667

STEP 9

Take the square root of the variance to find the population standard deviation.
σ=10.4266673.23 \sigma = \sqrt{10.426667} \approx 3.23

STEP 10

Round the population standard deviation to two decimal places.
σ3.23 \sigma \approx 3.23
The population standard deviation of the ratings for 2013-2014 is approximately 3.23 3.23 .

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