Question 1179666: A nationally known supermarket decided to promote its own brand of soft drinks on TV for two weeks. Before the ad campaign, the company randomly selected 21 of its stores across the United States to be part of a study to measure the campaign’s effectiveness. During a specified half-hour period on a certain Monday morning, all the stores in the sample counted the number of cans of its own brand of soft drink sold. After the campaign, a similar count was made. The average difference was an increase of 75 cans, with a standard deviation of difference of 30 cans. Using this information, construct a 90% confidence interval to estimate the population average difference in soft drink sales for this company’s brand before and after the ad campaign. Assume the differences in soft drink sales for the company’s brand are normally distributed in the population.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to construct the 90% confidence interval for the population average difference in soft drink sales:
**1. Identify the Given Information:**
* Sample size (n): 21 stores
* Sample mean difference (x̄_diff): 75 cans
* Sample standard deviation of differences (s_diff): 30 cans
* Confidence level: 90%
**2. Determine the Degrees of Freedom:**
* Degrees of freedom (df) = n - 1 = 21 - 1 = 20
**3. Find the Critical t-Value:**
* Since we have a small sample size (n < 30) and we're using the sample standard deviation, we'll use a t-distribution.
* For a 90% confidence level and 20 degrees of freedom, the critical t-value (t*) is approximately 1.725. You can find this value using a t-table or a calculator.
**4. Calculate the Standard Error:**
* Standard error (SE) = s_diff / √n = 30 / √21 ≈ 6.547
**5. Calculate the Margin of Error:**
* Margin of error (ME) = t* * SE = 1.725 * 6.547 ≈ 11.30
**6. Construct the Confidence Interval:**
* Lower bound = x̄_diff - ME = 75 - 11.30 ≈ 63.70
* Upper bound = x̄_diff + ME = 75 + 11.30 ≈ 86.30
**7. State the Confidence Interval:**
* The 90% confidence interval is approximately (63.70, 86.30).
**Interpretation:**
We are 90% confident that the true population average difference in soft drink sales (increase) after the ad campaign is between 63.70 cans and 86.30 cans.
|
|
|
