In a random sample of 100 customers, 62 returned with a month. For a 95% confidence interval on the proportion of customers who return for an oil change,
the lower limit is:
To calculate the lower limit for a 95% confidence interval on the proportion of customers who return for an oil change, you need to use the formula:
Lower Limit = Sample Proportion - Margin of Error
The first step is to calculate the sample proportion, which is simply the number of customers who returned for an oil change (62) divided by the total sample size (100):
Sample Proportion = Number of customers who returned for an oil change / Total sample size
Sample Proportion = 62 / 100 = 0.62
Next, you need to calculate the margin of error. The margin of error depends on the desired level of confidence and can be determined using a standard formula:
Margin of Error = Critical Value * Standard Error
For a 95% confidence interval, the critical value is approximately 1.96 (assuming a large sample size). The standard error can be calculated using the formula:
Standard Error = sqrt((Sample Proportion * (1 - Sample Proportion)) / Sample Size)
Plugging in the values:
Standard Error = sqrt((0.62 * (1 - 0.62)) / 100)
Standard Error ≈ 0.048
Finally, you can calculate the lower limit by subtracting the margin of error from the sample proportion:
Lower Limit = Sample Proportion - Margin of Error
Lower Limit = 0.62 - (1.96 * 0.048)
Lower Limit ≈ 0.525
Therefore, the lower limit for a 95% confidence interval on the proportion of customers who return for an oil change is approximately 0.525.