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.
136 82 199 174 97
170 222 115 120 169
164 93 132 171 0
93 0 93 110 440
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 95% confidence interval for the mean annual consumption of alcoholic beverages by European young women, we need to calculate the sample mean and the margin of error using the given data.
1. Calculate the sample mean (x̄):
- Add up all the values: 136 + 82 + 199 + 174 + 97 + 170 + 222 + 115 + 120 + 169 + 164 + 93 + 132 + 171 + 0 + 93 + 0 + 93 + 110 + 440 = 2666
- Divide the sum by the number of observations: x̄ = 2666 / 20 = 133.3
2. Calculate the standard deviation (s):
- Compute the differences between each observation and the sample mean. Square each difference.
- Sum up all the squared differences: (136 - 133.3)^2 + (82 - 133.3)^2 + ... + (110 - 133.3)^2 + (440 - 133.3)^2 = 441749.9
- Divide the sum by (n - 1), where n is the number of observations: s^2 = 441749.9 / (20 - 1) = 24652.8
- Take the square root of s^2 to get the standard deviation: s = sqrt(24652.8) ≈ 156.9
3. Calculate the margin of error (ME):
- The formula for the margin of error is: ME = t * (s / sqrt(n)), where t is the critical value and n is the sample size.
- Since the sample is small (n < 30) and the population is assumed symmetric, we can use the t-distribution.
- For a 95% confidence level and a sample size of 20, the critical value can be found using a t-table or calculator. It is approximately 2.093.
- Plug in the values to calculate the margin of error: ME = 2.093 * (156.9 / sqrt(20)) ≈ 92.6
4. Construct the confidence interval:
- The confidence interval is computed as: sample mean ± margin of error.
- Lower bound: x̄ - ME = 133.3 - 92.6 ≈ 40.7
- Upper bound: x̄ + ME = 133.3 + 92.6 ≈ 225.9
Therefore, the 95% confidence interval for the mean annual consumption of alcoholic beverages by European young women is approximately (40.7, 225.9) liters.