https://file.aiquickdraw.com/mathbot/file/1725192135725l8ntu3qg.jpg

Diketahui ada 600 kepala keluarga sebagai populasi yang membeli pakaian. Hasil penelitian terhadap 20 orang kepala keluarga ternyata jumlah pengeluaran untuk pakaian yang dibeli (ratusan ribuan rupiah) sbb.:

20

15

17

18

22

21

18

14

15

13

16

14

24

25

26

21

18

17

15

19

Hitunglah conf dence interval 95% rata-rata pengeluaran kepala keluarga tersebut

The slide in the image provides the formula for estimating the confidence interval for a population mean when the population variance is unknown and the sample size \( n < 30 \). In this case, we need to use the t-distribution to estimate the confidence interval.

### Given Data:

- Sample size, \( n = 20 \)
- Sample dataThe slide in the image provides the formula for estimating the confidence interval for a population mean when the population variance is unknown and the sample size \( n < 30 \). In this case, we need to use the t-distribution to estimate the confidence interval.

### Given Data:

- Sample size, \( n = 20 \)
- Sample  20, 15, 17, 18, 22, 21, 18, 14, 15, 13, 16, 14, 24, 25, 26, 21, 18, 17, 15, 19

### Steps to Calculate the 95% Confidence Interval:

1. **Calculate the sample mean \( \bar{x} \):**
   \[
   \bar{x} = \frac{\sum x_i}{n}
   \]
   
2. **Calculate the sample standard deviation \( S \):**
   \[
   S = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}
   \]

3. **Determine the t-score for the 95% confidence interval and \( n - 1 \) degrees of freedom:**
   - For \( n = 20 \), degrees of freedom \( = n - 1 = 19 \)
   - Use a t-distribution table or calculator to find \( t_{\alpha/2} \) for 95% confidence and 19 degrees of freedom.

4. **Calculate the margin of error:**
   \[
   \text{Margin of Error} = t_{\alpha/2} \times \frac{S}{\sqrt{n}}
   \]

5. **Determine the confidence interval:**
   \[
   \text{Confidence Interval} = \left( \bar{x} - \text{Margin of Error}, \bar{x} + \text{Margin of Error} \right)
   \]

Now, let's calculate these values.

### 1. Calculate the Sample Mean \( \bar{x} \)

\[
\bar{x} = \frac{20 + 15 + 17 + 18 + 22 + 21 + 18 + 14 + 15 + 13 + 16 + 14 + 24 + 25 + 26 + 21 + 18 + 17 + 15 + 19}{20}
\]

### 2. Calculate the Sample Standard Deviation \( S \)

\[
S = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}
\]

Let's calculate these values.### Calculations:

1. **Sample Mean \( \bar{x} \):**  
   \[
   \bar{x} = 18.4
   \]

2. **Sample Standard Deviation \( S \):**  
   \[
   S = 3.803
   \]

3. **t-score for 95% confidence and 19 degrees of freedom:**  
   \[
   t_{\alpha/2} = 2.093
   \]

4. **Margin of Error:**  
   \[
   \text{Margin of Error} = 1.780
   \]

5. **95% Confidence Interval for the Mean:**  
   \[
   (16.62, 20.18)
   \]

### Conclusion:

The 95% confidence interval for the average expenditure on clothing for the population of heads of families is between **16.62** and **20.18** hundred thousand rupiah.

Would you like more details on this calculation or have any questions?

Here are 5 related questions to consider:
1. How would the confidence interval change if the sample size were larger?
2. What is the impact of a higher confidence level (e.g., 99%) on the confidence interval width?
3. How is the t-distribution different from the normal distribution in statistical analysis?
4. How does the sample variance affect the width of the confidence interval?
5. What assumptions are made when calculating a confidence interval using the t-distribution?

**Tip:** Remember that as the sample size increases, the t-distribution approaches the normal distribution, which often leads to a narrower confidence interval for the same confidence level.

Learn how to calculate the 95% confidence interval for the mean expenditure on clothing for a population using statistical methods. This problem involves sample mean, sample standard deviation, margin of error, and the t-distribution theorem. Suitable for advanced high school or college-level students.

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