Consumption of alcoholic beverages by young women of drinking age has been increasing in the United Kingdom, the United States, and Europe (The Wall Street Journal, February 15, 2006). Data (annual consumption in liters) consistent with the findings reported in The Wall Street Journal article are shown for a sample of 20 European young women.
76 82 199 174 97
170 222 115 131 169
164 96 118 171 0
93 0 93 110 240
Assuming the population is roughly symmetric, construct a 95% confidence interval for the mean annual consumption of alcoholic beverages by European young women (to 1 decimal).
( , )
To construct a confidence interval for the mean annual consumption of alcoholic beverages by European young women, we will use the sample data provided and assume the population is roughly symmetric.
Step 1: Calculate the sample mean (x̄) and sample standard deviation (s).
x̄ = (76 + 82 + 199 + 174 + 97 + 170 + 222 + 115 + 131 + 169 + 164 + 96 + 118 + 171 + 0 + 93 + 0 + 93 + 110 + 240) / 20
= 1992 / 20
= 99.6
s = √((∑(xi - x̄)²) / (n - 1))
= √(((76 - 99.6)² + (82 - 99.6)² + ... + (240 - 99.6)²) / (20 - 1))
= √((48934.4) / 19)
≈ √2570.7579
≈ 50.7
Step 2: Determine the critical value for a 95% confidence interval.
Since the population is assumed to be approximately symmetric, we can use the t-distribution. With a sample size of 20 and a desired confidence level of 95%, we have (n - 1) = 19 degrees of freedom. Looking up the critical value in the t-distribution table or using a calculator, we find it to be approximately 2.093.
Step 3: Calculate the margin of error (E).
E = (critical value) * (standard deviation / √n)
= 2.093 * (50.7 / √20)
≈ 21.9
Step 4: Construct the confidence interval.
95% confidence interval = (sample mean - margin of error, sample mean + margin of error)
≈ (99.6 - 21.9, 99.6 + 21.9)
≈ (77.7, 121.5)
Therefore, the 95% confidence interval for the mean annual consumption of alcoholic beverages by European young women is approximately (77.7, 121.5) liters.
To construct a confidence interval for the mean annual consumption of alcoholic beverages by European young women, we can use the sample data provided.
Step 1: Calculate the sample mean.
To find the sample mean, add up all the observations and divide by the sample size.
Sample mean (x̄) = (76 + 82 + 199 + 174 + 97 + 170 + 222 + 115 + 131 + 169 + 164 + 96 + 118 + 171 + 0 + 93 + 0 + 93 + 110 + 240) / 20 = 137.8
Step 2: Calculate the sample standard deviation.
To find the sample standard deviation, we need to calculate the deviations from the sample mean, square them, sum them up, divide by n-1, and finally take the square root.
Deviation from mean = (x - x̄)
Squared deviation = (x - x̄)²
Sample variance (s²) = Σ(x - x̄)² / (n-1)
Sample standard deviation (s) = √(Σ(x - x̄)² / (n-1))
where Σ denotes sum and n is the sample size.
Using the provided data, we find:
Sample variance (s²) = (76-137.8)² + (82-137.8)² + ... + (240-137.8)² / (20-1)
= 68698.4 / 19 = 3615.7
Sample standard deviation (s) = √3615.7 ≈ 60.12
Step 3: Determine the critical value.
Since we are constructing a 95% confidence interval, we need to find the critical value associated with a 95% confidence level. For a symmetric population, we can use a t-distribution with n-1 degrees of freedom.
With a sample size of 20, the degrees of freedom is 20-1 = 19.
Using a t-distribution table or a statistical calculator, we find that the critical value for a 95% confidence level with 19 degrees of freedom is approximately t = 2.093.
Step 4: Calculate the margin of error.
The margin of error is calculated by multiplying the critical value with the standard deviation divided by the square root of the sample size.
Margin of error = t * (s / √n)
= 2.093 * (60.12 / √20) ≈ 26.47
Step 5: Construct the confidence interval.
The confidence interval is calculated by subtracting and adding the margin of error to the sample mean.
Confidence interval = (x̄ - Margin of error, x̄ + Margin of error)
= (137.8 - 26.47, 137.8 + 26.47)
= (111.33, 164.27)
Therefore, the 95% confidence interval for the mean annual consumption of alcoholic beverages by European young women is approximately (111.3, 164.3) liters.