Random samples were taken over 2 weeks (14 days) of a drink machine and data was collected of each drink sold. The sample mean was 65.2 drinks with a deviation of 8.1 drinks. Construct a 95% confidence interval on the mean number of drinks sold per day.
95% = mean ± 1.96 SEm
SEm = SD/√n
I'll let you do the calculations.
To construct a 95% confidence interval on the mean number of drinks sold per day, we will use the formula:
Confidence Interval = sample mean ± margin of error
First, we need to calculate the margin of error. The formula for the margin of error can be calculated as:
Margin of Error = (z-score) * (standard deviation / √n),
where:
- z-score is the critical value for the desired confidence level (we will use a 95% confidence level)
- standard deviation is the deviation of the sample mean
- n is the sample size
Since we have a sample size of 14 days, the sample size (n) is 14.
1. Find the z-score corresponding to the desired confidence level.
For a 95% confidence level, the z-score is 1.96. You can find this value from a standard normal distribution table or using statistical software.
2. Calculate the margin of error:
Margin of Error = (1.96) * (8.1 / √14)
3. Calculate the lower and upper bounds of the confidence interval:
Lower bound = sample mean - margin of error
Upper bound = sample mean + margin of error
Hence, the 95% confidence interval on the mean number of drinks sold per day is:
65.2 - Margin of Error < x < 65.2 + Margin of Error
Now, let's calculate the margin of error:
Margin of Error = (1.96) * (8.1 / √14)
= 3.602 (approximately)
Therefore, the 95% confidence interval on the mean number of drinks sold per day is:
65.2 - 3.602 < x < 65.2 + 3.602
Simplifying, the confidence interval is:
61.598 < x < 68.802
So, the 95% confidence interval on the mean number of drinks sold per day is approximately 61.598 to 68.802.