Electricity bills in a certain city have mean $108.73. Assume the bills are normally distributed with standard deviation $10.60. A sample of 47 bills was selected for an audit. Find the 46 percentile for the sample mean.
Write only a number as your answer. Round to two decimal places.
To find the 46th percentile for the sample mean, we can use the concept of the standard normal distribution.
First, we need to convert the sample mean to a z-score. The formula for calculating the z-score is:
z = (x - μ) / (σ / √n)
Where:
- x is the sample mean
- μ is the population mean
- σ is the population standard deviation
- n is the sample size
In this case, the population mean (μ) is given as $108.73, the population standard deviation (σ) is given as $10.60, and the sample size (n) is 47.
Let's substitute these values into the formula:
z = (x - μ) / (σ / √n)
z = (x - 108.73) / (10.60 / √47)
To find the 46th percentile, we need to find the z-score that corresponds to this percentile. We can use a standard normal distribution table or a calculator to do this. The z-score represents the number of standard deviations away from the mean.
Next, we need to find the z-score that corresponds to the 46th percentile. We can use a standard normal distribution table or a calculator for this. The z-score corresponding to a percentile can be interpreted as the value below which a given percentage of the data falls.
Once we have the z-score, we can use it to find the corresponding x-value (sample mean) by rearranging the formula:
x = z * (σ / √n) + μ
Finally, we can substitute the z-score into the formula to find the sample mean corresponding to the 46th percentile. Round the answer to two decimal places.