### (a) Calculate the Sample Mean and Sample Standard Deviation
To find the sample mean ( [tex]\(\bar{x}\)[/tex] ) and sample standard deviation ( [tex]\(s\)[/tex] ), we first need to list all the given prices:
[tex]\[
60, 5, 100, 110, 75, 95, 30, 23, 100, 110, 105, 5, 105, 60, 110, 120, 5, 90, 0, 70
\][/tex]
Sample Size ( [tex]\(n\)[/tex] ):
[tex]\[ n = 20 \][/tex]
Sample Mean ( [tex]\(\bar{x}\)[/tex] ):
The sample mean [tex]\(\bar{x}\)[/tex] is calculated by summing all the prices and then dividing by the number of prices:
[tex]\[
\bar{x} = \frac{60 + 5 + 100 + 110 + 75 + 95 + 30 + 23 + 100 + 110 + 105 + 5 + 105 + 60 + 110 + 120 + 5 + 90 + 0 + 70}{20} \][/tex]
Upon calculation, we get:
[tex]\[ \bar{x} = 68.9 \][/tex]
Sample Standard Deviation ( [tex]\(s\)[/tex] ):
The sample standard deviation [tex]\(s\)[/tex] is calculated using the formula for the sample standard deviation, which corrects for bias by using [tex]\( n-1 \)[/tex] in the denominator:
[tex]\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\][/tex]
Upon calculation, we find:
[tex]\[ s \approx 42.4015 \][/tex]
### (b) Find the 90% Confidence Interval for the Mean Price
To determine a 90% confidence interval for the mean price [tex]\( \mu \)[/tex], we use the t-distribution because the sample size is small ( [tex]\( n < 30 \)[/tex] ) and the population standard deviation is unknown.
Confidence Level ( [tex]\(\gamma\)[/tex] ):
[tex]\[ \gamma = 0.90 \][/tex]
This means the level of significance [tex]\( \alpha \)[/tex] is:
[tex]\[ \alpha = 1 - \gamma = 0.10 \][/tex]
Degrees of Freedom ( df ):
[tex]\[ df = n - 1 = 20 - 1 = 19 \][/tex]
Using the t-distribution table, for a 90% confidence level and 19 degrees of freedom, the t-critical value [tex]\(t_{ \text{critical} } \)[/tex] is approximately:
[tex]\[ t_{ \text{critical} } \approx 1.729 \][/tex]
Next, we calculate the margin of error ( [tex]\( E \)[/tex] ):
[tex]\[
E = t_{ \text{critical} } \times \left( \frac{s}{\sqrt{n}} \right)
\][/tex]
[tex]\[
E = 1.729 \times \left( \frac{42.4015}{\sqrt{20}} \right)
\][/tex]
Upon calculation, we find:
[tex]\[ E \approx 16.39 \][/tex]
Finally, the confidence interval for the mean price [tex]\(\mu\)[/tex]:
[tex]\[
(\bar{x} - E) \text{ to } (\bar{x} + E)
\][/tex]
[tex]\[
68.9 - 16.39 \text{ to } 68.9 + 16.39
\][/tex]
[tex]\[
52.51 \text{ to } 85.29
\][/tex]
### Conclusion
(a) The sample mean price [tex]\(\bar{x}\)[/tex] is [tex]$68.90 and the sample standard deviation \(s\) is approximately $[/tex]42.4015.
(b) The 90% confidence interval for the mean price of all sleeping bags in the given temperature range is:
52.51 to 85.29 dollars.
