Given:

The 95% confidence interval of the average alcohol content is (7.8, 9.6)

Sample size \((n) = 50\)

a) Confidence level decreased to 90%

Using z tables, the critical value for two tailed 95% confidence interval is 1.96 or 90% is 1.645 If critical value decreases, the margin of error also decreases.

Let's understand with an example.

Suppose the sample mean is 4.

Standard error of mean is 2.

Computing confidence intervals

\(\overline{x}-z_{\frac{a}{2}}\times SE, \overline{x}+z_{\frac{a}{2}}\times SE\)

\(95\% : 4 - 1 1.96 \times 2, 4 + 1.96 \times 2\)

\(: 0.08, 7.92\)

\(90\% : 4 - 1.645 \times 2, 4 + 1.645 \times 2\)

\(, 0.71, 7.29\)

Width: upper limit - lower limit

\(95\% : 7.92 - 0.08 = 7.84\)

\(90\% : 7.29 - 0.71 = 6.58\)

Now, it could be seen that the width decreased when the confidence level is decreases. So, 90% confidence interval will be narrower as z critical value decreases.

b) Interpretation of confidence interval

1. If suppose, 100 confidence intervals are constructed with the sample size, then 95 of them would contain the true population parameter.

2. We are 95% confident that the true population parameter lies with the confidence interval. In confidence interval, the interval varies but the population parameter remains fixed.

Therefore, we are 95% confident that the population mean is within 7.8, 9.6 and 5% that it is not.

c)As mentioned earlier, in confidence interval, the population parameter remains fixed and the interval changes.

It does not gives any idea of the number of samples containing the true population parameter. It gives an opinion on the overall population parameter.

Therefore, it is not a correct statement.

The 95% confidence interval of the average alcohol content is (7.8, 9.6)

Sample size \((n) = 50\)

a) Confidence level decreased to 90%

Using z tables, the critical value for two tailed 95% confidence interval is 1.96 or 90% is 1.645 If critical value decreases, the margin of error also decreases.

Let's understand with an example.

Suppose the sample mean is 4.

Standard error of mean is 2.

Computing confidence intervals

\(\overline{x}-z_{\frac{a}{2}}\times SE, \overline{x}+z_{\frac{a}{2}}\times SE\)

\(95\% : 4 - 1 1.96 \times 2, 4 + 1.96 \times 2\)

\(: 0.08, 7.92\)

\(90\% : 4 - 1.645 \times 2, 4 + 1.645 \times 2\)

\(, 0.71, 7.29\)

Width: upper limit - lower limit

\(95\% : 7.92 - 0.08 = 7.84\)

\(90\% : 7.29 - 0.71 = 6.58\)

Now, it could be seen that the width decreased when the confidence level is decreases. So, 90% confidence interval will be narrower as z critical value decreases.

b) Interpretation of confidence interval

1. If suppose, 100 confidence intervals are constructed with the sample size, then 95 of them would contain the true population parameter.

2. We are 95% confident that the true population parameter lies with the confidence interval. In confidence interval, the interval varies but the population parameter remains fixed.

Therefore, we are 95% confident that the population mean is within 7.8, 9.6 and 5% that it is not.

c)As mentioned earlier, in confidence interval, the population parameter remains fixed and the interval changes.

It does not gives any idea of the number of samples containing the true population parameter. It gives an opinion on the overall population parameter.

Therefore, it is not a correct statement.