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 survey information provided.
The survey states that in a randomly selected sample of n = 935 American adults, 39% said they do not get enough sleep each night. The margin of error for the 95% confidence level is reported to be 3.3%.
To calculate the confidence interval, we'll need to consider the sample proportion (p-hat) and the margin of error. The sample proportion is given as 39% (0.39), and the margin of error is 3.3% (0.033).
First, we calculate the standard error using the formula:
Standard Error = sqrt((p-hat * (1 - p-hat)) / n)
Standard Error = sqrt((0.39 * (1 - 0.39)) / 935)
Next, we calculate the margin of error using the formula:
Margin of Error = Critical value * Standard Error
For a 95% confidence level, the critical value is approximately 1.96.
Margin of Error = 1.96 * Standard Error
Now, we can calculate the lower and upper bounds of the confidence interval:
Lower bound = p-hat - Margin of Error
Upper bound = p-hat + Margin of Error
Lower bound = 0.39 - (1.96 * Standard Error)
Upper bound = 0.39 + (1.96 * Standard Error)
By plugging the values into these equations, we can find the answers.
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.