Sleep Awareness Week begins in the spring with the release of the National Sleep Foundation’s annual poll of U.S. sleep habits and ends with the beginning of daylight savings time, when most people lose an hour of sleep. In the foundation’s random sample of 1029 U.S. adults, 48% reported that they often or always” got enough sleep during the last 7 nights.
- Identify the parameter of interest.
- Check if the conditions for constructing a confidence interval for p are met.
- Find the critical value for a 98% confidence interval. Then calculate the interval.
- Interpret the interval in context.
The Correct Answer and Explanation is :
1. Parameter of Interest
The parameter of interest is the true proportion (pp) of all U.S. adults who often or always got enough sleep during the last 7 nights.
2. Conditions for Constructing a Confidence Interval
To construct a confidence interval for pp, the following conditions must be satisfied:
- Random Sample: The sample of 1029 U.S. adults is stated to be random, so this condition is met.
- Independence: The sample size n=1029n = 1029 is less than 10% of the U.S. adult population, ensuring independence.
- Large Enough Sample Size: The sample size must be large enough such that np≥10np \geq 10 and n(1−p)≥10n(1-p) \geq 10.
- np=1029×0.48=493.92≥10np = 1029 \times 0.48 = 493.92 \geq 10
- n(1−p)=1029×0.52=535.08≥10n(1-p) = 1029 \times 0.52 = 535.08 \geq 10
All conditions are satisfied.
3. Critical Value and Confidence Interval
For a 98% confidence level, the critical value (z∗z^*) can be found using a standard normal table or calculator: z∗=2.33z^* = 2.33
The formula for the confidence interval is: p^±z∗p^(1−p^)n\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1 – \hat{p})}{n}}
Substituting the values: p^=0.48,n=1029\hat{p} = 0.48, \quad n = 1029 Margin of Error=z∗p^(1−p^)n=2.330.48×0.521029≈0.0355\text{Margin of Error} = z^* \sqrt{\frac{\hat{p}(1 – \hat{p})}{n}} = 2.33 \sqrt{\frac{0.48 \times 0.52}{1029}} \approx 0.0355
Confidence interval: 0.48±0.0355 ⟹ (0.4445,0.5155)0.48 \pm 0.0355 \implies (0.4445, 0.5155)
4. Interpretation
We are 98% confident that the true proportion of U.S. adults who often or always got enough sleep during the last 7 nights is between 44.45% and 51.55%. This means that if we were to take many random samples of the same size and construct confidence intervals, approximately 98% of those intervals would contain the true proportion.