The average amount customers at a certain grocery store spend yearly is $784.39. Assume the variable is normally distributed. If the standard deviation is $76.15, find the probability that a randomly selected customer spends between $712.05 and $932.88.
To find the probability that a randomly selected customer spends between $712.05 and $932.88, we need to standardize the values using the formula for z-score:
z = (x - mean) / standard deviation
Let's calculate the z-scores for $712.05 and $932.88:
For $712.05:
z1 = ($712.05 - $784.39) / $76.15
For $932.88:
z2 = ($932.88 - $784.39) / $76.15
Now, we can look up the corresponding probabilities in the standard normal distribution table. The cumulative probability for z1 represents the probability of a customer spending less than or equal to $712.05, and the cumulative probability for z2 represents the probability of a customer spending less than or equal to $932.88. To find the probability between the two values, we subtract the cumulative probability of z1 from the cumulative probability of z2:
P($712.05 < x < $932.88) = P(z1 < z < z2) = P(z < z2) - P(z < z1)
You can use a standard normal distribution table or a statistical calculator with a normal distribution function to find the cumulative probabilities for z1 and z2, and then calculate the difference between them.