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.
209 82 199 174 97
170 222 115 130 169
164 97 118 171 0
93 0 93 110 127
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).
95% = mean ± 1.96 SEm
SEm = SD/√n
Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
I'll let you do the calculations.
To construct a confidence interval for the mean annual consumption of alcoholic beverages by European young women, you can follow these steps:
Step 1: Calculate the sample mean.
Compute the mean of the given data set by summing up all the values and dividing it by the sample size (20 in this case).
(209 + 82 + 199 + 174 + 97 + 170 + 222 + 115 + 130 + 169 + 164 + 97 + 118 + 171 + 0 + 93 + 0 + 93 + 110 + 127) / 20 = 1411 / 20 = 70.55
Step 2: Calculate the sample standard deviation.
Determine the standard deviation of the data set using the formula for sample standard deviation.
s = sqrt([(Σx^2) / n] - [(Σx / n)^2])
First, calculate the sum of the squares of the data values (x^2).
Then, square the sum of the data values (∑x) divided by the sample size (n).
Subtract the second calculation from the first.
Finally, take the square root of the result.
For simplicity, we can use statistical software or calculators to obtain the sample standard deviation. In this case, the sample standard deviation is approximately 62.1.
Step 3: Determine the critical value.
Since the population is assumed to be symmetric, we can utilize the t-distribution with degrees of freedom (df) equal to the sample size minus 1 (20 - 1 = 19) to find the critical value associated with a 95% confidence level.
Let's refer to a t-table or use statistical software to obtain the critical value.
At a 95% confidence level and 19 degrees of freedom, the critical value is approximately 2.093.
Step 4: Calculate the margin of error.
The margin of error represents the range within which the true population mean is likely to fall.
To compute the margin of error, multiply the critical value by the sample standard deviation divided by the square root of the sample size.
Margin of error = (Critical value) * (Sample standard deviation / √(Sample size))
Margin of error = 2.093 * (62.1 / √20) ≈ 27.78
Step 5: Construct the confidence interval.
Finally, construct the confidence interval by adding and subtracting the margin of error from the sample mean.
The lower bound is the sample mean minus the margin of error, and the upper bound is the sample mean plus the margin of error.
Lower bound = Sample mean - Margin of error
Upper bound = Sample mean + Margin of error
Lower bound = 70.55 - 27.78 ≈ 42.8
Upper bound = 70.55 + 27.78 ≈ 98.3
Therefore, the 95% confidence interval for the mean annual consumption of alcoholic beverages by European young women is approximately (42.8, 98.3) liters.