To construct a 80% confidence interval (CI) for the mean of a population, based on a sample of 18 values with a mean of 287.2 and a standard deviation of 18, follow these steps:
1. Determine the sample size [tex]\(n\)[/tex]:
[tex]\[
n = 18
\][/tex]
2. Calculate the degrees of freedom [tex]\(df\)[/tex]:
[tex]\[
df = n - 1 = 18 - 1 = 17
\][/tex]
3. Determine the confidence level [tex]\( \alpha \)[/tex]:
The confidence level is 80%, or 0.80. Since this is a two-tailed test, you divide the remaining probability by 2:
[tex]\[
\frac{1 + 0.80}{2} = 0.90
\][/tex]
4. Find the critical value [tex]\( t_c \)[/tex] for [tex]\( \alpha = 0.80 \)[/tex]:
Using a t-distribution table, or a calculator with t-distribution functionality, find the t-value corresponding to [tex]\( \alpha = 0.10 \)[/tex] in each tail with [tex]\(df = 17\)[/tex]:
[tex]\[
t_c \approx 1.33
\][/tex]
5. Calculate the standard error (SE):
[tex]\[
SE = \frac{\text{sample standard deviation}}{\sqrt{n}} = \frac{18}{\sqrt{18}} \approx \frac{18}{4.24} \approx 4.24
\][/tex]
6. Compute the margin of error (ME):
[tex]\[
ME = t_c \times SE = 1.33 \times 4.24 \approx 5.64
\][/tex]
7. Calculate the lower and upper bounds of the confidence interval:
[tex]\[
\text{Lower bound} = \text{sample mean} - ME = 287.2 - 5.64 \approx 281.56
\][/tex]
[tex]\[
\text{Upper bound} = \text{sample mean} + ME = 287.2 + 5.64 \approx 292.84
\][/tex]
8. Construct the confidence interval:
[tex]\[
(281.56, 292.84)
\][/tex]
So, the 80% confidence interval for the mean of the population is:
[tex]\[
(281.56, 292.84)
\][/tex]
