What is the age distribution of promotion$-$sensitive shoppers? A supermarket super shopper is defined as a shopper for whom at least $70\%$ of the items purchased were on sale or purchased with a coupon.

Age range, years $18-2829-3940-5051-6162$ and over

Midpoint x $2334455667$

Percent of super shoppers $10\%47\%21\%9\%13\%$

A button hyperlink to the SALT program that reads: Use SALT.

For the $62-$and$-$over group, use the midpoint $67$ years.

$($a$)$

Using the age midpoints x and the percentage of super shoppers, do we have a valid probability distribution? Explain.

Yes. The events are indistinct and the probabilities sum to less than $1.$

Yes. The events are distinct and the probabilities do not sum to $1.$

No$.$ The events are indistinct and the probabilities sum to more than $1.$

No$.$ The events are distinct and the probabilities sum to $1.$

Yes. The events are distinct and the probabilities sum to $1.$

$($b$)$

Use a histogram to graph the probability distribution of part $($a$).($Because the data table has summarized the data into categories, use SALT to create a bar chart.$)$

A graph has a horizontal axis labeled Age and a vertical axis labeled $\%$ Super Shoppers. The histogram has $5$ bars. Each bar is associated with a label and an approximate value as listed below.

$23$: $47$

$34$: $13$

$45$: $10$

$56$: $21$

$67$: $9$

A graph has a horizontal axis labeled Age and a vertical axis labeled $\%$ Super Shoppers. The histogram has $5$ bars. Each bar is associated with a label and an approximate value as listed below.

$23$: $10$

$34$: $47$

$45$: $21$

$56$: $9$

$67$: $13$

A graph has a horizontal axis labeled Age and a vertical axis labeled $\%$ Super Shoppers. The histogram has $5$ bars. Each bar is associated with a label and an approximate value as listed below.

$23$: $13$

$34$: $9$

$45$: $21$

$56$: $47$

$67$: $10$

A graph has a horizontal axis labeled Age and a vertical axis labeled $\%$ Super Shoppers. The histogram has $5$ bars. Each bar is associated with a label and an approximate value as listed below.

$23$: $13$

$34$: $21$

$45$: $10$

$56$: $9$

$67$: $47$

$($c$)$

Compute the expected age $$ of a super shopper. $($Round your answer to two decimal places.$)$

$=$ yr

$($d$)$

Compute the standard deviation $$ for ages of super shoppers. $($Round your answer to two decimal places.$)$

$=$ yr