The number of cars sold annually by used car salespeople is normally distributed with a standard deviation of 18. A random sample of 310 salespeople was taken and the mean number of cars sold annually was found to be 71. Find the 94% confidence interval estimate of the population mean.
To find the 94% confidence interval estimate of the population mean, we can follow these steps:
Step 1: Identify the given information
- Sample size (n): 310
- Sample mean (x̄): 71
- Standard deviation (σ): 18
- Confidence level: 94%
Step 2: Calculate the margin of error
The margin of error is calculated using the formula:
Margin of error = (Z-value) * (Standard deviation / √n)
To find the Z-value corresponding to a 94% confidence level, we need to find the Z-value that leaves 3% in each tail. Because the normal distribution is symmetric, we can subtract (100% - 94% = 6%) from one tail and divide it by 2. So, we have (6% / 2 = 3%).
Using a Z-table or a statistical calculator, we can find the Z-value corresponding to a cumulative probability of 0.9700 (1 - 0.03) = 1.8808.
Now, we can substitute the values into the formula:
Margin of error = 1.8808 * (18 / √310) ≈ 2.1099
Step 3: Calculate the confidence interval
The confidence interval is calculated as:
Confidence interval = (Sample mean) ± (Margin of Error)
Substituting the values we have:
Confidence interval = 71 ± 2.1099
Therefore, the 94% confidence interval estimate of the population mean is approximately 68.8901 to 73.1099.
94% = mean ± 1.88 SEm
SEm = SD/√n