The age of customers for a particular retail store follows a normal distribution with a mean of 37.5 years and standard deviation of 15 years given the sample size is 36
A compute standard error?
B what is the probability that the next customer who enters the store will be more than 31 years old?
C what is the probability that the next customer who enters the store will be less than 42 years old?
A) Standard error (SE) = standard deviation / square root of sample size
SE = 15 / sqrt(36)
SE = 2.5
B) To find the probability of a customer being more than 31 years old, we need to convert this value to a z-score using the formula:
z = (x - mean) / standard error
z = (31 - 37.5) / 2.5
z = -2.6
Using a z-score table or calculator, we can find that the probability of a z-score being less than -2.6 is 0.0047. Therefore, the probability of the next customer being more than 31 years old is approximately 1 - 0.0047 = 0.9953 (or 99.53%).
C) To find the probability of a customer being less than 42 years old, we again need to convert this value to a z-score:
z = (42 - 37.5) / 2.5
z = 1.8
Using a z-score table or calculator, we can find that the probability of a z-score being less than 1.8 is 0.9641. Therefore, the probability of the next customer being less than 42 years old is 0.9641 (or 96.41%).
To compute the standard error, you need to use the formula:
Standard Error = Standard Deviation / √(Sample Size)
Given the standard deviation is 15 years and the sample size is 36, let's calculate the standard error:
A) Standard Error = 15 / √36
Standard Error = 15 / 6
Standard Error = 2.5 years
Therefore, the standard error is 2.5 years.
To find the probability that the next customer who enters the store will be more than 31 years old, we need to calculate the z-score and then look up the corresponding probability from the standard normal distribution table or use a calculator.
B) Z-score = (Observed Value - Mean) / Standard Deviation
Z-score = (31 - 37.5) / 15
Z-score = -6.5 / 15
Z-score = -0.4333
Using the standard normal distribution table or a calculator, we can find the probability:
Probability = 1 - (Area to the left of the z-score)
From the standard normal distribution table or calculator, the area to the left of the z-score -0.4333 is 0.3336. Subtracting this from 1 gives us:
Probability = 1 - 0.3336
Probability = 0.6664
Therefore, the probability that the next customer who enters the store will be more than 31 years old is approximately 0.6664 or 66.64%.
To find the probability that the next customer who enters the store will be less than 42 years old, we again need to calculate the z-score and then look up the corresponding probability.
C) Z-score = (Observed Value - Mean) / Standard Deviation
Z-score = (42 - 37.5) / 15
Z-score = 4.5 / 15
Z-score = 0.3
Using the standard normal distribution table or a calculator, we can find the probability:
Probability = Area to the left of the z-score
From the standard normal distribution table or calculator, the area to the left of the z-score 0.3 is 0.6179.
Therefore, the probability that the next customer who enters the store will be less than 42 years old is approximately 0.6179 or 61.79%.