To construct a 90% confidence interval for the mean amount of sleep based on the given data, we can use the following steps:
- Calculate the Sample Mean (\( \bar{x} \)).
- Calculate the Sample Standard Deviation (\( s \)).
- Determine the t-value for 90% confidence with \( n - 1 \) degrees of freedom.
- Calculate the Standard Error (SE).
- Construct the Confidence Interval.
Step 1: Calculate the Sample Mean (\( \bar{x} \))
The sample data is:
- \( 8.5, 7.5, 6.5, 9, 6, 8.5, 8 \)
\[ \bar{x} = \frac{8.5 + 7.5 + 6.5 + 9 + 6 + 8.5 + 8}{7} = \frac{54.5}{7} \approx 7.7857 \]
Step 2: Calculate the Sample Standard Deviation (\( s \))
First, we find the deviations from the mean, square them, and then take the average to find the variance.
\[ \text{Deviations:} (8.5 - 7.7857)^2, (7.5 - 7.7857)^2, (6.5 - 7.7857)^2, (9 - 7.7857)^2, (6 - 7.7857)^2, (8.5 - 7.7857)^2, (8 - 7.7857)^2 \] Calculating the squared deviations:
\[ (0.7143)^2 = 0.5101 \] \[ (-0.2857)^2 = 0.0816 \] \[ (-1.2857)^2 = 1.6525 \] \[ (1.2143)^2 = 1.4775 \] \[ (-1.7857)^2 = 3.1965 \] \[ (0.7143)^2 = 0.5101 \] \[ (0.2143)^2 = 0.0458 \]
Now we find the variance:
\[ \text{Variance} = \frac{\sum \text{deviations}^2}{n - 1} = \frac{0.5101 + 0.0816 + 1.6525 + 1.4775 + 3.1965 + 0.5101 + 0.0458}{7 - 1} \approx \frac{7.4732}{6} \approx 1.2455 \]
Then the standard deviation:
\[ s = \sqrt{1.2455} \approx 1.1166 \]
Step 3: Determine the t-value for 90% confidence
Degrees of freedom (\(df\)) = \(n - 1 = 8 - 1 = 7\).
Using a t-table, the t-value for 90% confidence with 7 degrees of freedom is approximately \( t \approx 1.895 \).
Step 4: Calculate Standard Error (SE)
\[ SE = \frac{s}{\sqrt{n}} = \frac{1.1166}{\sqrt{8}} \approx \frac{1.1166}{2.8284} \approx 0.3947 \]
Step 5: Construct the Confidence Interval
The confidence interval is given by:
\[ \bar{x} \pm t \cdot SE \]
Calculating the interval:
\[ 7.7857 \pm 1.895 \cdot 0.3947 \] Calculating \( t \cdot SE \):
\[ 1.895 \cdot 0.3947 \approx 0.748 \]
Finally, the confidence interval:
\[ (7.7857 - 0.748, 7.7857 + 0.748) \approx (7.0377, 8.5337) \]
Rounding to two decimal places:
\[ \text{Confidence Interval} \approx (7.04, 8.53) \]
Thus, the 90% confidence interval for the mean amount of sleep is (7.04, 8.53).