Question

The masses of 6 containers of Chow dog food are:

9.8

10.4

9.8

10

10.2

9.6





















































































Find a 90% confidence interval for the mean mass of all such dog food. (Assume an approximate normal distribution).

Answer 1

To calculate a 90% confidence interval for the mean mass of all such dog food, we can follow these steps:

### Step 1: Calculate the sample mean (\(\bar{x}\)) and sample standard deviation (s)

1. **Sample Mean (\(\bar{x}\))**:
\[
\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}
\]
where \(n\) is the number of samples (6 in this case).

\[
\bar{x} = \frac{9.8 + 10.4 + 9.8 + 10 + 10.2 + 9.6}{6} = \frac{60.8}{6} \approx 10.13
\]

2. **Sample Standard Deviation (s)**:
\[
s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}}
\]

First, we need to calculate each \((x_i - \bar{x})^2\):
- \((9.8 - 10.13)^2 \approx 0.1089\)
- \((10.4 - 10.13)^2 \approx 0.0729\)
- \((9.8 - 10.13)^2 \approx 0.1089\)
- \((10 - 10.13)^2 \approx 0.0169\)
- \((10.2 - 10.13)^2 \approx 0.0049\)
- \((9.6 - 10.13)^2 \approx 0.2809\)

Now sum these squared deviations:
\[
\sum (x_i - \bar{x})^2 \approx 0.1089 + 0.0729 + 0.1089 + 0.0169 + 0.0049 + 0.2809 \approx 0.5924
\]

Now calculate the sample standard deviation:
\[
s = \sqrt{\frac{0.5924}{6 - 1}} = \sqrt{\frac{0.5924}{5}} \approx \sqrt{0.11848} \approx 0.344
\]

### Step 2: Determine the critical value

For a 90% confidence interval and \(n-1 = 5\) degrees of freedom, we look up the t-distribution:
- The critical t-value (\(t_{0.05, 5}\)) for a 90% confidence interval with 5 degrees of freedom is approximately \(t_{0.05, 5} = 2.015\).

### Step 3: Calculate the margin of error (E)

The margin of error is calculated as:
\[
E = t \cdot \left(\frac{s}{\sqrt{n}}\right)
\]
where \(s\) is the sample standard deviation, and \(n\) is the sample size.

\[
E = 2.015 \cdot \left(\frac{0.344}{\sqrt{6}}\right) \approx 2.015 \cdot \left(\frac{0.344}{2.449}\right) \approx 2.015 \cdot 0.140 = 0.2821
\]

### Step 4: Construct the confidence interval

The confidence interval is given by:
\[
\bar{x} \pm E
\]
\[
CI = [10.13 - 0.2821, 10.13 + 0.2821]
\]
\[
CI \approx [9.8479, 10.4121]
\]

### Conclusion

The 90% confidence interval for the mean mass of all such dog food is approximately:

**\[ (9.85, 10.41) \]** (rounded to two decimal places).