Question

the accompanying table describes probabilities for the california daily 4 lottery. the player selects four digits with repetition allowed, and the random variable x is the number of digits that match those in the same order that they are drawn (for a \straight\ bet). use the range rule of thumb to determine whether 4 matches is a significantly high number of matches.
select the correct choice below and, if necessary, fill in the answer box within your choice
○ a. significantly high numbers of matches are or less. since 4 is greater than this value, 4 matches is not a significantly high number of matches.
(round to one decimal place as needed.)
○ b. significantly high numbers of matches are or more. since 4 is at least as high as this value, 4 matches is a significantly high number of matches.
(round to one decimal place as needed.)
○ c. significantly high numbers of matches are or more. since 4 is less than this value, 4 matches is not a significantly high number of matches.
(round to one decimal place as needed.)
○ d. significantly high numbers of matches are or less. since 4 is at least as low as this value, 4 matches is a significantly high number of matches.
(round to one decimal place as needed.)
○ e. not enough information is given.

Explanation:

Step1: Calculate the mean \(\mu\)

The formula for the mean of a discrete probability distribution is \(\mu=\sum xP(x)\).

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Step2: Calculate the variance \(\sigma^{2}\)

The formula for the variance is \(\sigma^{2}=\sum(x - \mu)^{2}P(x)\)

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Step3: Calculate the standard deviation \(\sigma\)

\(\sigma=\sqrt{\sigma^{2}}=\sqrt{0.3694}\approx0.6\)

Step4: Use the range rule of thumb

The range rule of thumb states that values \(x\) are significantly high if \(x\geq\mu + 2\sigma\)
\(\mu+2\sigma=0.406+2\times0.6=0.406 + 1.2=1.6\)

Answer:

B. Significantly high numbers of matches are \(1.6\) or more. Since \(4\) is at least as high as this value, \(4\) matches is a significantly high number of matches.