In a survey, 39% of a randomly selected sample of n = 935 American adults said that they do not get enough sleep each night. The margin of error for the 95% confidence level was reported to be 3.3%.
(a) Use the survey information to create a 95% confidence interval for the percentage that feels they don't get enough sleep every night.
___% to ___%
Write a sentence that interprets this interval. Specify the population.
With 95% confidence, we can say that in the population of
American adults, between ____% and ___% would say they don't get enough sleep each night.
To create a 95% confidence interval for the percentage of American adults who feel they don't get enough sleep each night, we can use the following steps:
Step 1: Calculate the standard error.
To estimate the population proportion, we can use the formula:
SE = sqrt[(p * (1 - p)) / n]
Where:
- p is the proportion of the sample (39% or 0.39).
- n is the sample size (935).
Plugging in the values, we get:
SE = sqrt[(0.39 * (1 - 0.39)) / 935] ≈ 0.0143
Step 2: Calculate the margin of error.
The margin of error is given as 3.3% (0.033). We can calculate it by multiplying the standard error by the z-value corresponding to a 95% confidence level.
For a 95% confidence level, the z-value is approximately 1.96 (you can obtain this value from a standard normal distribution table or using statistical software).
Margin of Error = z * SE
Margin of Error = 1.96 * 0.0143 ≈ 0.028
Step 3: Calculate the confidence interval.
The confidence interval can be calculated by subtracting the margin of error from the sample proportion and adding it to the sample proportion.
Lower Limit = Sample Proportion - Margin of Error
Lower Limit = 0.39 - 0.028 ≈ 0.362
Upper Limit = Sample Proportion + Margin of Error
Upper Limit = 0.39 + 0.028 ≈ 0.418
Therefore, the 95% confidence interval for the percentage of American adults who feel they don't get enough sleep each night is approximately 36.2% to 41.8%.
With 95% confidence, we can say that in the population of American adults, between 36.2% and 41.8% would say they don't get enough sleep each night.