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).
( , )
Find mean and standard deviation, then use confidence interval formula.
CI95 = mean ± 1.96(sd/√n)
n = 20
I'll let you take it from here.
To construct a 95% confidence interval for the mean annual consumption of alcoholic beverages by European young women, we can use the following formula:
Confidence Interval = sample mean ± (critical value) × (standard deviation / square root of sample size)
Step 1: Calculate the sample mean.
Add up all the values and divide by the number of observations:
76 + 82 + 199 + 174 + 97 + 170 + 222 + 115 + 131 + 169 + 164 + 96 + 118 + 171 + 0 + 93 + 0 + 93 + 110 + 240 = 2678 (sum of values)
2678 / 20 = 133.9 (sample mean)
Step 2: Calculate the standard deviation.
Subtract the sample mean from each observation, square the result, sum up the squared differences, divide by the number of observations minus 1, and take the square root:
((76 - 133.9)^2 + (82 - 133.9)^2 + (199 - 133.9)^2 + (174 - 133.9)^2 + (97 - 133.9)^2 + (170 - 133.9)^2 + (222 - 133.9)^2 + (115 - 133.9)^2 + (131 - 133.9)^2 + (169 - 133.9)^2 + (164 - 133.9)^2 + (96 - 133.9)^2 + (118 - 133.9)^2 + (171 - 133.9)^2 + (0 - 133.9)^2 + (93 - 133.9)^2 + (0 - 133.9)^2 + (93 - 133.9)^2 + (110 - 133.9)^2 + (240 - 133.9)^2) = 866968.2 (sum of squared differences)
866968.2 / (20 - 1) = 45630.9579 (variance)
√(45630.9579) = 213.7832 (standard deviation)
Step 3: Determine the critical value.
We need to find the critical value from the t-distribution table with a confidence level of 95% and degrees of freedom (df) equal to the sample size minus 1.
Since the sample size is 20, the df = 20 - 1 = 19.
Looking up a t-distribution table, the critical value for a 95% confidence level and 19 degrees of freedom is approximately 2.093.
Step 4: Calculate the margin of error.
Multiply the critical value by the standard deviation divided by the square root of the sample size:
2.093 × (213.7832 / √(20)) = 100.275 (margin of error)
Step 5: Construct the confidence interval.
The confidence interval is calculated by subtracting and adding the margin of error to the sample mean:
Lower limit = sample mean - margin of error
Upper limit = sample mean + margin of error
Lower limit = 133.9 - 100.275 = 33.625
Upper limit = 133.9 + 100.275 = 234.175
Therefore, the 95% confidence interval for the mean annual consumption of alcoholic beverages by European young women is (33.6, 234.2) liters.