Hello please help..
The owner of a computer repair shop has determined that their daily revenue has mean $7200 and standard deviation $1200. The daily revenue totals for the next 30 days will be monitored. What is the probability that the mean daily revenue for the next 30 days will be between $7000 and $7500? Round to four decimal places
To find the probability that the mean daily revenue for the next 30 days will be between $7000 and $7500, we need to use the concept of the sampling distribution of the sample mean.
The sampling distribution of the sample mean is approximately normally distributed if the sample size is large enough. In this case, since we are given the population mean and standard deviation, and the sample size is 30 (which is generally considered large enough), we can assume that the sampling distribution of the sample mean will be approximately normal.
To calculate the probability, we need to standardize the mean daily revenue values of $7000 and $7500 into z-scores.
The formula for the z-score is: z = (x - μ) / (σ / sqrt(n))
Where:
x = the value we want to standardize
μ = population mean
σ = population standard deviation
n = sample size
For $7000:
z_7000 = (7000 - 7200) / (1200 / sqrt(30))
For $7500:
z_7500 = (7500 - 7200) / (1200 / sqrt(30))
Next, we need to look up the probability corresponding to each z-score in the standard normal distribution table, or use a calculator or statistical software to find the area under the standard normal curve between these z-scores.
Once we determine the z-scores and corresponding probabilities, we can subtract the probability corresponding to the lower z-score from the probability corresponding to the higher z-score to find the probability that the mean daily revenue will be between $7000 and $7500.
Note: If the sample size is small (less than 30), we would use the t-distribution instead of the standard normal distribution for the calculations. But in this case, with a sample size of 30, the normal distribution approximation is acceptable.
Please calculate the values for z_7000 and z_7500, and lookup the corresponding probabilities in the standard normal distribution table or using a calculator or statistical software, and then subtract the probability of z_7000 from the probability of z_7500 to find the desired probability.
You can play around with Z table stuff at
http://davidmlane.com/hyperstat/z_table.html