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 exceed $7000?
Broooo put the answer aswell pleasee!
To find the probability that the mean daily revenue for the next 30 days will exceed $7000, we can use the Central Limit Theorem.
The Central Limit Theorem states that for a large sample size, the distribution of the sample mean will be approximately normally distributed, regardless of the shape of the original population distribution.
To calculate this probability, we first need to calculate the standard error of the mean (SEM) using the formula:
SEM = standard deviation / √ sample size
In this case, the sample size is 30, and the standard deviation is $1200. So, the SEM is:
SEM = $1200 / √30
Next, we need to calculate the z-score using the formula:
z = (sample mean - population mean) / SEM
The population mean is $7200, and we want to know the probability of the sample mean exceeding $7000. So, the sample mean is $7000. Plugging the values into the formula, we get:
z = ($7000 - $7200) / SEM
Now, we can use a standard normal distribution table or a calculator to find the probability associated with the calculated z-score.
The probability is the area under the standard normal distribution curve to the right of the z-score. This represents the probability that the mean daily revenue for the next 30 days will exceed $7000.
Let's calculate the z-score and find the corresponding probability using the formula and a standard normal distribution table or calculator.
To find the probability that the mean daily revenue for the next 30 days will exceed $7000, we can use the Central Limit Theorem in statistics. The Central Limit Theorem states that the distribution of sample means will be approximately normally distributed regardless of the shape of the original population, as long as the sample size is large enough.
In this case, since the owner has already calculated the mean and standard deviation of the daily revenue, we can assume that they have a population of revenue data.
To find the probability, we need to standardize the sample mean using the standard deviation of the population. The formula for standardizing a sample mean is:
Z = (X - μ) / (σ / sqrt(n))
Where:
Z = the z-score
X = the value we want to standardize (in this case $7000)
μ = the population mean
σ = the population standard deviation
n = the sample size
In this case, the population mean (μ) is $7200, the population standard deviation (σ) is $1200, and the sample size (n) is 30. Plugging in the values:
Z = (7000 - 7200) / (1200 / sqrt(30))
Now, we can use a standard normal distribution table or a statistical calculator to find the probability associated with the calculated z-score. Assuming a normal distribution, we can find the probability using the cumulative distribution function (CDF) at this z-score.
The steps to find the probability using a standard normal distribution table are as follows:
1. Calculate the z-score: Z = (7000 - 7200) / (1200 / sqrt(30)) = -2.041
2. Look up the corresponding area in the standard normal distribution table for the z-score of -2.041. The table shows the area to the left of the z-score, and to find the area to the right, subtract the value from 1. In this case, the area to the left of -2.041 is 0.0205. So, the area to the right (probability that mean daily revenue exceeds $7000) is 1 - 0.0205 = 0.9795.
Therefore, the probability that the mean daily revenue for the next 30 days will exceed $7000 is approximately 0.9795, or 97.95%.
Z = (mean1 - mean2)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.