Questions: It has been conjectured by the U.S. Census Bureau that "approximately 60% of foreign-born people who live in the U.S. are not naturalized citizens". Suppose that in a national random sample of 70 foreign-born people who live in the U.S., on average, how many people would you expect to get that are not naturalized citizens. Select the best answer below. 28 42 48 50 None of the above

It has been conjectured by the U.S. Census Bureau that "approximately 60% of foreign-born people who live in the U.S. are not naturalized citizens". Suppose that in a national random sample of 70 foreign-born people who live in the U.S., on average, how many people would you expect to get that are not naturalized citizens. Select the best answer below.
28
42
48
50
None of the above

Transcript text: 4) It has been conjectured by the U.S. Census Bureau that "approximately $60 \%$ of foreign-born people who live in the U.S. are not naturalized citizens". Suppose that in a national random sample of 70 foreign-born people who live in the U.S., on average, how many people would you expect to get that are not naturalized citizens. Select the best answer below. 28 42 48 50 None of the above

Solution

Solution Steps

Step 1: Define the Problem

We are tasked with determining the expected number of foreign-born individuals in a random sample of 70 who are not naturalized citizens, given that approximately $60\%$ of foreign-born people in the U.S. are not naturalized citizens.

Step 2: Identify Parameters

Let:

$n = 70$ (the number of trials, or foreign-born individuals sampled)
$p = 0.60$ (the probability of an individual not being a naturalized citizen)
$q = 1 - p = 0.40$ (the probability of an individual being a naturalized citizen)

Step 3: Calculate the Mean

The expected number of non-naturalized citizens can be calculated using the formula for the mean of a binomial distribution: \[ \mu = n \cdot p \] Substituting the values: \[ \mu = 70 \cdot 0.60 = 42.0 \]

Step 4: Calculate Variance and Standard Deviation

The variance and standard deviation can also be calculated for additional context:

Variance: \[ \sigma^2 = n \cdot p \cdot q = 70 \cdot 0.60 \cdot 0.40 = 16.8 \]
Standard Deviation: \[ \sigma = \sqrt{n \cdot p \cdot q} = \sqrt{70 \cdot 0.60 \cdot 0.40} \approx 4.1 \]

Step 5: Summarize Results

The expected number of non-naturalized citizens in the sample of 70 foreign-born individuals is $42.0$.