Question

1. Suppose that 24% of U.S. residents are fluent in a language other than English. Let X represent the number of people in a SRS of 12 U.S. residents who are fluent in another language.

1. (3 pts) What distribution will X have? (Give the name of this type of distribution.)

b. (4 pts) Find the mean and standard deviation of the distribution of X.

3. (6 pts) Calculate the probability that a random sample of 12 U.S. residents contains fewer than 3 who are fluent in a language other than English.

Answer 1

Part 1

X ~ Bin(n = 12, p = 0.24) [ANSWER]

Explanation:

We are given that 24% of U.S. residents are fluent in a language other than English. Thus, the probability that a randomly selected U.S. resident is fluent in a language other than English is 24% = 0.24.

Now, we are given that a S.R.S. of 12 U.S. residents is taken and X denotes the number of people who are fluent in a language other than English in this sample.

Since, the number of U.S. residents is fixed (fixed number of trials), each U.S. resident has two outcomes (fluent or not fluent in a language other than English) and each U.S. resident is fluent in a language other than English independent of other U.S. residents in the sample. Thus, we can conclude that:

X ~ Bin(n = 12, p = 0.24)

Part 2

The mean of X is given by:

The standard deviation of X is given by:

Part 3

Since, X ~ Bin(n = 12, p = 0.24) the probability mass function of X is given by:

The probability that a random sample of 12 U.S. residents contains fewer than 3 who are fluent in a language other than English is given by:

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