Answer:
The 90% confidence interval for the mean number of energy drinks consumed per week is (3.44, 3.56).
Explanation:
Confidence Interval for the Mean Number of Energy Drinks Consumed
Given:
Sample size (n) = 1083
Sample mean (x) = 3.5
Population standard deviation (σ) = 1.3
Confidence level (1 - α) = 90%
Step 1: Find the critical value (z)
For a 90% confidence interval, the α level is 1 - 0.90 = 0.10.
Since the confidence interval is symmetrical, we split the α level into two equal parts, each with an area of 0.05.
Using a z-table or calculator, we find the z-score that corresponds to an area of 0.95 to the right (1 - 0.05).
This value is approximately 1.645.
Step 2: Calculate the standard error of the mean
The standard error of the mean (SEM) is calculated as follows:
SEM = σ / sqrt(n) = 1.3 / sqrt(1083) ≈ 0.038
Step 3: Calculate the margin of error
The margin of error (ME) is calculated as follows:
ME = z * SEM = 1.645 * 0.038 ≈ 0.063
Step 4: Construct the confidence interval
The confidence interval is constructed by adding and subtracting the margin of error from the sample mean:
Lower bound = x - ME = 3.5 - 0.063 ≈ 3.44
Upper bound = x + ME = 3.5 + 0.063 ≈ 3.56
Therefore, the 90% confidence interval for the mean number of energy drinks consumed per week is (3.44, 3.56).
Interpretation:
We are 90% confident that the true population mean number of energy drinks consumed per week lies between 3.44 and 3.56.
Question:
A market research company wishes to know how many energy drinks adults drink each week. They want to construct a 90% confidence interval for the mean and are assuming that the population standard deviation for the number of energy drinks consumed each week is 1.3. The study found that for a sample of 1083 adults the mean number of energy drinks consumed per week is 3.5. Construct the desired confidence interval. Round your answers to one decimal place.