Suppose 100 randomly selected customers were asked to rate a particular service they recently had on a scale of 0 to 10. The sample mean rating was found to be 7.3. (Assume the population standard deviation of all ratings in the population of customers is 0.8.) Using all this information, estimate the average rating of this service for all customers.
To estimate the average rating of the service for all customers, we can use the concept of confidence intervals.
A confidence interval provides a range of values within which we can expect the true population mean to fall, based on a sample mean and a measure of variability. In this case, we will use the sample mean rating of 7.3 and the population standard deviation of 0.8.
To calculate the confidence interval, we need to determine the level of confidence we want. For example, if we want a 95% confidence interval, we would use a standard deviation multiplier of 1.96 (which corresponds to a 95% confidence level for a large sample size).
The formula to calculate the confidence interval is:
Confidence Interval = Sample Mean ± (Standard Deviation Multiplier) * (Standard Deviation / √Sample Size)
In this case, the sample size is 100, the sample mean is 7.3, the standard deviation multiplier for a 95% confidence level is 1.96, and the population standard deviation is 0.8.
Plugging these values into the formula, we get:
Confidence Interval = 7.3 ± (1.96) * (0.8 / √100)
Simplifying the equation, we have:
Confidence Interval = 7.3 ± (1.96) * (0.8 / 10)
Calculating this further, we get:
Confidence Interval ≈ 7.3 ± 0.1568
Therefore, the 95% confidence interval for the average rating of the service for all customers is approximately (7.14, 7.46). This means that we can be 95% confident that the true average rating of the service falls within this range.