Question

Suppose it is known that the distribution of purchase amounts by customers entering a popular retail store is approximately normal with mean $100 and standard deviation $10. What is the probability that a randomly selected customer spends less than $105 at this store?

Answer 1

Answer:

The probability that a randomly selected customer spends less than $105 at this store is 0.6915 or 69.15%

Step-by-step explanation:

1. Let's review the information given to us to answer the question correctly:

Mean of the purchase amounts by customers entering a popular retail store = $ 100

Standard deviation of the purchase amounts by customers entering a popular retail store = $ 10

2. What is the probability that a randomly selected customer spends less than $105 at this store?

For calculating the probability, we need to find the z-score first, this way:

X = 105

z-score = (X - μ)/σ

Replacing with the values we have:

z-score = 105 - 100/10

z-score = 5/10 = 0.5

Now, let's calculate the probability, using the z-table:

p (z = 0.5) = 0.6915

The probability that a randomly selected customer spends less than $105 at this store is 0.6915 or 69.15%

Answer 2

Answer:

69.15% probability that a randomly selected customer spends less than $105 at this store

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

$\mu = 100, \sigma = 10$

What is the probability that a randomly selected customer spends less than $105 at this store?

This is the pvalue of Z when X = 105. So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{105 - 100}{10}$

$Z = 0.5$

$Z = 0.5$ has a pvalue of 0.6915

69.15% probability that a randomly selected customer spends less than $105 at this store