Answered

Select ALL the correct answers.

Ten students at a local college were randomly selected and asked how many hours they spend studying on the weekend. The data collected is shown in the table below.

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Student & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
\begin{tabular}{c}
Hours Spent \\
Studying
\end{tabular} & 4 & 6 & 1 & 0 & 5 & 8 & 2 & 3 & 3 & 4 \\
\hline
\end{tabular}

The formulas below are provided for reference, where [tex]$E$[/tex] is the margin of error, [tex]$\sigma$[/tex] is the population standard deviation, and [tex]$n$[/tex] is the sample size.

\begin{tabular}{|c|c|}
\hline
\begin{tabular}{c}
Margin of Error \\
(95\% Confidence)
\end{tabular} & \begin{tabular}{c}
Minimum Sample Size Needed \\
[tex]$(95 \%$[/tex] Confidence [tex]$)$[/tex]
\end{tabular} \\
\hline
[tex]$E=1.96 \cdot \frac{\sigma}{\sqrt{n}}$[/tex] & [tex]$n=\left(\frac{1.96 \cdot \sigma}{E}\right)^2$[/tex] \\
\hline
\end{tabular}

Assuming the data for the population is normally distributed with a standard deviation of 2.4 hours, which of the statements are true?

1. For a [tex]$95 \%$[/tex] confidence interval, the approximate margin of error is 2.231 hours.
2. The [tex]$95 \%$[/tex] confidence interval of the population mean is from 2.112 to 5.088 hours.
3. The estimated population mean is 4 hours.
4. For a [tex]$95 \%$[/tex] confidence interval, the approximate margin of error is 1.488 hours.



Answer :

GinnyAnswer
To solve this question, we will follow a structured approach involving the calculation of the sample mean, the margin of error for a 95% confidence interval, and the confidence interval itself.

### Step-by-Step Solution:

#### 1. Calculate the Sample Mean
First, we find the average number of hours the students studied.

- The data provided is: [4, 6, 1, 0, 5, 8, 2, 3, 3, 4]
- The number of observations (n) is 10.
- Sum of hours: 4 + 6 + 1 + 0 + 5 + 8 + 2 + 3 + 3 + 4 = 36

The sample mean ( [tex]\(\bar{x}\)[/tex] ) is calculated as:
[tex]\[ \bar{x} = \frac{\text{Sum of hours}}{n} = \frac{36}{10} = 3.6 \][/tex]

#### 2. Calculate the Margin of Error
The margin of error (E) for a 95% confidence interval is calculated using the formula:
[tex]\[ E = 1.96 \cdot \frac{\sigma}{\sqrt{n}} \][/tex]
Given:
- Population standard deviation ([tex]\(\sigma\)[/tex]) = 2.4
- Sample size (n) = 10

Plugging in the values, we get:
[tex]\[ E = 1.96 \cdot \frac{2.4}{\sqrt{10}} \][/tex]
[tex]\[ E = 1.96 \cdot \frac{2.4}{3.162} \approx 1.488 \][/tex]

#### 3. Calculate the 95% Confidence Interval
The confidence interval is given by:
[tex]\[ \text{Confidence Interval} = \left( \bar{x} - E, \bar{x} + E \right) \][/tex]
Using the sample mean ([tex]\(\bar{x} = 3.6\)[/tex]) and the margin of error (E = 1.488), we calculate:
[tex]\[ \text{Lower bound} = 3.6 - 1.488 = 2.112 \][/tex]
[tex]\[ \text{Upper bound} = 3.6 + 1.488 = 5.088 \][/tex]
Thus, the 95% confidence interval is:
[tex]\[ (2.112, 5.088) \][/tex]

### Summary of Findings:
- The sample mean is calculated to be 3.6.
- The margin of error is 1.488.
- The 95% confidence interval is from 2.112 to 5.088.

### Checking the Statements:
1. For a 95% confidence interval, the approximate margin of error is 2.231 hours.
- This is incorrect because the margin of error was found to be 1.488 hours.

2. The 95% confidence interval of the population mean is from 2.112 to 5.088 hours.
- This is correct, as calculated above.

3. The estimated population mean is 4 hours.
- This is incorrect because the sample mean, which estimates the population mean, is 3.6 hours.

4. For a 95% confidence interval, the approximate margin of error is 1.488 hours.
- This is correct, as calculated above.

### Correct Statements:
- The 95% confidence interval of the population mean is from 2.112 to 5.088 hours.
- For a 95% confidence interval, the approximate margin of error is 1.488 hours.

Thus, the correct statements are:
- (2.112, 5.088)
- 1.488

Other Questions