Question

the table below gives the number of hours spent watching tv last week by a sample of 24 children.

\

$$\begin{array}{|c|c|c|c|c|} \\hline 9 & 4 & 1 & 1 & 8 \\\\ \\hline 4 & 10 & 9 & 9 & 9 \\\\ \\hline 7 & 4 & 3 & 4 & 3 \\\\ \\hline 4 & 2 & 5 & 1 & 9 \\\\ \\hline 3 & 8 & 8 & 10 & \\\\ \\hline \\end{array}$$

range:
standard deviation:

please round your answers to 4 decimal places.

Explanation:

Identify given data

We are given a sample of \(n = 24\) observations representing the number of hours spent watching TV last week by 24 children:

$$ \{9, 4, 1, 1, 8, 4, 10, 9, 9, 9, 7, 4, 3, 4, 3, 4, 2, 5, 1, 9, 3, 8, 8, 10\} $$

Calculate the range

The range is the difference between the maximum and minimum values in the dataset.

$$ \text{Maximum value} = 10 $$

$$ \text{Minimum value} = 1 $$

$$ \text{Range} = 10 - 1 = 9 $$

Calculate the sample mean

First, find the sum of all observations, \(\sum x_i\):

$$ \sum x_i = 9+4+1+1+8+4+10+9+9+9+7+4+3+4+3+4+2+5+1+9+3+8+8+10 = 126 $$

Now, calculate the sample mean, \(\bar{x}\):

$$ \bar{x} = \frac{\sum x_i}{n} = \frac{126}{24} = 5.25 $$

Calculate the sum of squared differences

Next, calculate \(\sum (x_i - \bar{x})^2\):

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Let's count the frequencies of each value to simplify the calculation:

• Value 1: frequency = 3
• Value 2: frequency = 1
• Value 3: frequency = 3
• Value 4: frequency = 5
• Value 5: frequency = 1
• Value 7: frequency = 1
• Value 8: frequency = 3
• Value 9: frequency = 5
• Value 10: frequency = 2

Total count check: \(3 + 1 + 3 + 5 + 1 + 1 + 3 + 5 + 2 = 24\).
Now calculate the sum of squared deviations:

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Calculate the sample standard deviation

Since this is a sample of 24 children, we use the sample standard deviation formula:

$$ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} $$

$$ s = \sqrt{\frac{229}{24 - 1}} = \sqrt{\frac{229}{23}} $$

$$ s = \sqrt{9.956521739} \approx 3.155395655 $$

Rounding to 4 decimal places gives \(3.1554\).

Answer:

Range: 9
Standard Deviation: 3.1554