(a) Using the income midpoints x and the percent of super shoppers, do we have a valid probability distribution? Explain. Yes. The events are indistinct and the probabilities sum to less than 1.No. The events are indistinct and the probabilities sum to 1. No. The events are indistinct and the probabilities sum to more than 1.Yes. The events are distinct and the probabilities do not 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). (Select the correct graph.) (c) Compute the expected income ? of a super shopper (in thousands of dollars). (Enter a number. Round your answer to two decimal places.) ? = thousands of dollars (d) Compute the standard deviation ? for the income of super shoppers (in thousands of dollars). (Enter a number. Round your answer to two decimal places.) ? = thousands of dollars
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
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What is the income distribution of super 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. In the following table, income units are in thousands of dollars, and each interval goes up to but does not include the given high value. The midpoints are given to the nearest thousand dollars.
| Income |
5-15 | 15-25 | 25-35 | 35-45 | 45-55 | 55 or more |
|---|---|---|---|---|---|---|
| Midpoint x | 10 | 20 | 30 | 40 | 50 | 60 |
| Percent of super shoppers | 22% | 14% | 20% | 17% | 19% | 8% |
(a)
Using the income midpoints x and the percent of super shoppers, do we have a valid(b)
Use a histogram to graph the probability distribution of part (a). (Select the correct graph.)(c)
Compute the expected income ? of a super shopper (in thousands of dollars). (Enter a number. Round your answer to two decimal places.)? = thousands of dollars
(d)
Compute the standard deviation ? for the income of super shoppers (in thousands of dollars). (Enter a number. Round your answer to two decimal places.)? = thousands of dollars
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