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).
( , )
n=20, xbar= 126,
s=63.94
a=0.05, |t(0.025, df=n-1=19)|= 2.09
So 95% CI is
xbar ± t*s/√n
126 ± 2.09*63.94/√(20)
(96.1, 155.9)
You're welcome.
Thanks for all your help, I really appreciate all the time you took to help me out. And thanks for showing me step by step.
To construct a confidence interval for the mean annual consumption of alcoholic beverages by European young women, we can use the following formula:
Confidence Interval = sample mean ± (t-value * standard error)
1. Calculate the sample mean:
To find the sample mean, add up all the values and divide by the sample size:
Sample Mean = (76 + 82 + 199 + 174 + 97 + 170 + 222 + 115 + 131 + 169 + 164 + 96 + 118 + 171 + 0 + 93 + 0 + 93 + 110 + 240) / 20
Sample Mean = 1913 / 20
Sample Mean = 95.65 (rounded to two decimal places)
2. Calculate the standard error:
Standard Error = sample standard deviation / square root of sample size
First, we need to find the sample standard deviation by calculating the sum of the squared differences between each observation and the sample mean, then dividing by (n - 1). After finding the sample standard deviation, we can calculate the standard error.
Sample Standard Deviation = √(Σ(xi - x̄)² / (n - 1))
Squared differences: (76 - 95.65)², (82 - 95.65)², ..., (0 - 95.65)², (93 - 95.65)², (110 - 95.65)², (240 - 95.65)²
Sum of squared differences = (76 - 95.65)² + (82 - 95.65)² + ... + (0 - 95.65)² + (93 - 95.65)² + (110 - 95.65)² + (240 - 95.65)²
Standard Deviation = √((Sum of squared differences) / (n - 1))
Standard Deviation = √(80643.75 / 19)
Standard Deviation = √4249.197368421052
Standard Deviation = 65.16 (rounded to two decimal places)
Standard Error = 65.16 / √20
Standard Error = 14.55 (rounded to two decimal places)
3. Find the t-value:
A 95% confidence level corresponds to a t-value with a significance level of 0.05 (two-tailed) and degrees of freedom (df) equal to the sample size minus 1 (df = 20 - 1 = 19). We can use a t-distribution table or a calculator to find the t-value.
For a 95% confidence interval and 19 degrees of freedom, the t-value is approximately 2.093.
4. Calculate the confidence interval:
Confidence Interval = 95.65 ± (2.093 * 14.55)
Confidence Interval = 95.65 ± 30.41
Confidence Interval = (65.24, 125.06) (rounded to one decimal place)
Therefore, the 95% confidence interval for the mean annual consumption of alcoholic beverages by European young women is (65.2, 125.1) liters.