historically, the average customer spends $8.53 for all TGP restaurants(u=8.53). If the population standard deviation is $1.06(o=1.06) and the data is normally distributed, find the probabiltiy that the sample mean will be within $.15 of the populaiton mean using a sample size of 64 customers?
The sample mean will be within $.15 of the population mean using a sample size of 121 customers?
The sample mean will be between $8.30 and $8.68 using a sample size of 121 customers?
no idea
To find the probability in these scenarios, we can use the concept of the standard normal distribution. Since the data is normally distributed and we know the population standard deviation, we can use the Z-score formula.
The Z-score formula is given by: Z = (X - μ) / (σ / √n)
Where:
Z is the Z-score,
X is the sample mean,
μ is the population mean,
σ is the population standard deviation, and
n is the sample size.
We can then use the Z-score to look up the corresponding probability from the standard normal distribution table or use a statistical calculator.
Now let's calculate the probabilities for each scenario:
1. The sample mean will be within $0.15 of the population mean using a sample size of 64 customers.
Here, X = μ ± $0.15, n = 64, μ = $8.53, and σ = $1.06.
Calculate the Z-score for the upper and lower limits:
Lower Z-score (Z_l) = ($8.53 - $0.15 - $8.53) / ($1.06 / √64)
Upper Z-score (Z_u) = ($8.53 + $0.15 - $8.53) / ($1.06 / √64)
Calculate the probability using the Z-scores:
P(Z_l < Z < Z_u) = P(Z_u) - P(Z_l)
2. The sample mean will be within $0.15 of the population mean using a sample size of 121 customers.
Here, X = μ ± $0.15, n = 121, μ = $8.53, and σ = $1.06.
Calculate the Z-score for the upper and lower limits:
Lower Z-score (Z_l) = ($8.53 - $0.15 - $8.53) / ($1.06 / √121)
Upper Z-score (Z_u) = ($8.53 + $0.15 - $8.53) / ($1.06 / √121)
Calculate the probability using the Z-scores:
P(Z_l < Z < Z_u) = P(Z_u) - P(Z_l)
3. The sample mean will be between $8.30 and $8.68 using a sample size of 121 customers.
Here, X = $8.30, n = 121, μ = $8.53, and σ = $1.06.
Calculate the Z-score for the lower and upper limits:
Lower Z-score (Z_l) = ($8.30 - $8.53) / ($1.06 / √121)
Upper Z-score (Z_u) = ($8.68 - $8.53) / ($1.06 / √121)
Calculate the probability using the Z-scores:
P(Z_l < Z < Z_u) = P(Z_u) - P(Z_l)
To find the corresponding probabilities, you can refer to the standard normal distribution table or use statistical software/tools to solve these equations.