https://file.aiquickdraw.com/mathbot/file/1725192308161t2gw3wos.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 formula shown in the image is the confidence interval for the population mean when the population standard deviation is unknown, and the sample size is small. The formula is given by:

\[
\bar{x} - t_{\frac{\alpha}{2}, n-1} \frac{s}{\sqrt{n}} < \mu < \bar{x} + t_{\frac{\alpha}{2}, n-1} \frac{s}{\sqrt{n}}
\]

Where:
- \(\bar{x}\) = sample mean
- \(s\) = sample standard deviation
- \(n\) = sample size
- \(t_{\frac{\alpha}{2}, n-1}\) = critical value from the t-distribution with \(n-1\) degrees of freedom
- \(\alpha\) = significance level (for a 95% confidence interval, \(\alpha = 0.05\))

Let's calculate the 95% confidence interval for the mean expenditure.

### Step 1: Calculate the sample mean (\(\bar{x}\))

\[
\bar{x} = \frac{\sum x_i}{n}
\]

### Step 2: Calculate the sample standard deviation (\(s\))

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

### Step 3: Find the t-critical value for 95% confidence interval

For 95% confidence and \(n-1 = 19\) degrees of freedom, we will find the \(t\)-critical value.

Let's compute these steps:Here's the calculation for the 95% confidence interval of the average expenditure of the households:

- **Sample Mean (\(\bar{x}\))**: 18.4 (hundreds of thousands of rupiahs)
- **Sample Standard Deviation (\(s\))**: 3.80
- **t-Critical Value (\(t_{\frac{\alpha}{2}, 19}\))**: 2.093 (for 19 degrees of freedom at 95% confidence level)

The 95% confidence interval for the mean expenditure is:

\[
(16.62, 20.18)
\]

This means we are 95% confident that the true average expenditure for the entire population of households lies between 16.62 and 20.18 hundred thousand rupiahs.

Would you like more details on any of these steps or have any other questions?

Here are some related questions you might be interested in:

1. How is the critical value \(t_{\frac{\alpha}{2}, n-1}\) determined from the t-distribution table?
2. What happens to the confidence interval if the sample size increases?
3. How would this calculation change if the population standard deviation was known?
4. What is the impact of changing the confidence level on the confidence interval width?
5. How do we interpret a 95% confidence interval in practical terms?

**Tip:** A larger sample size generally results in a narrower confidence interval, providing a more precise estimate of the population mean.

Learn how to calculate the 95% confidence interval for the mean expenditure of households based on a sample. Includes steps for finding sample mean, sample standard deviation, and critical t-value. Get a detailed explanation and interpretation.

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