Questions: According to a poll, 46% of people in a country 18 years old or older stated that they had read at least six books (fiction and nonfiction) within the past year. You conduct a random sample of 250 people from that country 18 years old or older. Complete parts (a) through (e) below. (b) Use the normal distribution to approximate the probability that exactly 125 read at least six books within the past year. Interpret this result. The probability is . (Round to four decimal places as needed.) Interpret this result. Select the correct choice below and fill in (Round to the nearest whole number as needed.) A. In a sample of 250 adults, approximately will be expected to state that they have read at least 6 books within the past year. B. Out of every 100 samples of 250 adults, approximately will result in exactly 125 who state that they have read at least 6 books within the past year. (c) Use the normal distribution to approximate the probability that fewer than 120 read at least six books within the past year. Interpret this result. The probability is . (Round to four decimal places as needed.) Interpret this result. Select the correct choice below and fill in the answer box to complete your choice.

Solution Steps

To find the probability that exactly 125 out of 250 adults read at least six books, we use the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

where:

• \( n = 250 \) (sample size),
• \( x = 125 \) (number of successes),
• \( p = 0.46 \) (probability of success),
• \( q = 1 - p = 0.54 \) (probability of failure).

Calculating this gives:

\[ P(X = 125) = 0.0226 \]

Next, we calculate the mean, variance, and standard deviation of the binomial distribution:

• Mean \( \mu = n \cdot p = 250 \cdot 0.46 = 115.0 \)
• Variance \( \sigma^2 = n \cdot p \cdot q = 250 \cdot 0.46 \cdot 0.54 = 62.1 \)
• Standard Deviation \( \sigma = \sqrt{npq} = \sqrt{250 \cdot 0.46 \cdot 0.54} = 7.8804 \)

The interpretation of the probability result is as follows:

Out of every 100 samples of 250 adults, approximately \( 2 \) will result in exactly \( 125 \) who state that they have read at least \( 6 \) books within the past year.

To find the probability that fewer than 120 adults read at least six books, we approximate this using the normal distribution. We calculate the Z-scores for the range:

\[ P(X < 120) = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(0.6345) - \Phi(-\infty) = 0.7371 \]

The interpretation of the probability result is as follows:

In a sample of 250 adults, approximately \( 74 \) will be expected to state that they have read fewer than \( 6 \) books within the past year.

Final Answer