https://file.aiquickdraw.com/mathbot/file/1725192453656fxmz9g8g.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    Menggunakan rumus tersebut beserta cara penyelesaian nya

To solve the problem, we need to calculate the 95% confidence interval for the average spending of the 20 households on clothing. Here's how we can approach the problem:

### Step 1: Summarize the Data
The spending values provided are:

20, 15, 17, 18, 22, 21, 18, 14, 15, 13, 16, 14, 24, 25, 26, 21, 18, 17, 15, 19

Let's first calculate the sample mean \(\bar{x}\) and sample standard deviation \(S\).

### Step 2: Calculate the Sample Mean \(\bar{x}\)
The sample mean is calculated as:

\[
\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i
\]

where \(n = 20\) is the number of samples, and \(x_i\) are the individual spending values.

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

### Step 3: Calculate the Sample Standard Deviation \(S\)
The sample standard deviation \(S\) is calculated using the formula:

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

### Step 4: Determine the Critical Value \(t_{\alpha/2}\)
Since the population variance is unknown and the sample size is small (\(n < 30\)), we use the \(t\)-distribution. For a 95% confidence interval and \(n-1 = 19\) degrees of freedom, the critical value \(t_{\alpha/2}\) can be found from \(t\)-distribution tables.

### Step 5: Calculate the Confidence Interval
The confidence interval for the mean is given by:

\[
\bar{x} - t_{\alpha/2} \times \frac{S}{\sqrt{n}} < \mu < \bar{x} + t_{\alpha/2} \times \frac{S}{\sqrt{n}}
\]

Let's perform the calculations.### Results

1. **Sample Mean (\(\bar{x}\))**: The mean spending is \(18.4\) (in hundreds of thousands of Rupiah).
2. **Sample Standard Deviation (\(S\))**: The standard deviation is approximately \(3.80\).
3. **Critical Value (\(t_{\alpha/2}\))**: The critical value for a 95% confidence level with 19 degrees of freedom is approximately \(2.093\).
4. **95% Confidence Interval**: The confidence interval for the mean spending is approximately \((16.62, 20.18)\).

### Interpretation
We are 95% confident that the true average spending of the entire population of 600 households on clothing lies between **16.62** and **20.18** (in hundreds of thousands of Rupiah).

---

Do you want more details or have any questions?

Here are some related questions:
1. How would the confidence interval change with a larger sample size?
2. What happens if the confidence level is changed to 99%?
3. Can we use the z-distribution instead of the t-distribution? Why or why not?
4. What if the data were not normally distributed—how would that affect our results?
5. How would you calculate the confidence interval if the population variance was known?

**Tip:** Understanding when to use the t-distribution versus the z-distribution is key in statistical inference, especially for small sample sizes or unknown population variances.

Learn how to calculate the 95% confidence interval for the average spending of 20 households on clothing. Includes step-by-step calculation of sample mean, sample standard deviation, critical value determination using t-distribution, and interpretation. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation