What is the distribution of \( X \)?
Distribution of \( X \)
The distribution of \( X \) is given by \( X \sim N(7.17, 1.72) \).
\(\boxed{X \sim N(7.17, 1.72)}\)
What is the distribution of \( \sum x \)?
Distribution of \( \sum x \)
The distribution of \( \sum x \) is given by \( \sum x \sim N(358.5, 12.1622) \).
\(\boxed{\sum x \sim N(358.5, 12.1622)}\)
What is the probability that one randomly selected customer will spend more than \( 7.3552 \)?
Calculation of probability
The probability that one randomly selected customer will spend more than \( 7.3552 \) is \( P(X > 7.3552) = 1 - P(X \leq 7.3552) = 0.4571 \).
\(\boxed{0.4571}\)
For the 50 customers, find the probability that their average spent is less than \( 7.3452 \).
Calculation of average probability
The probability that the average spent by 50 customers is less than \( 7.3452 \) is \( P(\bar{X} < 7.3452) = 0.7643 \).
\(\boxed{0.7643}\)
Find the probability that the randomly selected 50 customers will spend more than \( 367.2600 \).
Calculation of total spending probability
The probability that the 50 customers will spend more than \( 367.2600 \) is \( P(\sum x > 367.2600) = 0.2357 \).
\(\boxed{0.2357}\)
Is the assumption of normal necessary?
Assumption of normality
The assumption of normality is necessary for the calculations made in this context.
\(\boxed{\text{Yes}}\)
The owner of Don Mcalds gives a coupon for a free sundae to the 4, at least how much must a group of 50 spend in total to get the free sundae?
Minimum total spending calculation
The minimum amount a group of 50 must spend to get a free sundae is \( 375.5888 \).
\(\boxed{375.5888}\)
The distribution of \( X \) is \( \boxed{X \sim N(7.17, 1.72)} \).
The distribution of \( \sum x \) is \( \boxed{\sum x \sim N(358.5, 12.1622)} \).
The probability that one randomly selected customer will spend more than \( 7.3552 \) is \( \boxed{0.4571} \).
The probability that the average spent by 50 customers is less than \( 7.3452 \) is \( \boxed{0.7643} \).
The probability that the 50 customers will spend more than \( 367.2600 \) is \( \boxed{0.2357} \).
The assumption of normality is necessary, \( \boxed{\text{Yes}} \).
The minimum amount a group of 50 must spend to get a free sundae is \( \boxed{375.5888} \).