Questions: Use a normal approximation to find the probability of the indicated number of voters. In this case, assume that 158 eligible voters aged 18-24 are randomly selected. Suppose a previous study showed that among eligible voters aged 18-24, 22% of them voted. Probability that fewer than 37 voted The probability that fewer than 37 of 158 eligible voters voted is (Round to four decimal places as needed.)

Use a normal approximation to find the probability of the indicated number of voters. In this case, assume that 158 eligible voters aged 18-24 are randomly selected. Suppose a previous study showed that among eligible voters aged 18-24, 22% of them voted.

Probability that fewer than 37 voted

The probability that fewer than 37 of 158 eligible voters voted is
(Round to four decimal places as needed.)

Transcript text: Use a normal approximation to find the probability of the indicated number of voters. In this case, assume that 158 eligible voters aged $18-24$ are randomly selected. Suppose a previous study showed that among eligible voters aged $18-24,22 \%$ of them voted. Probability that fewer than 37 voted The probability that fewer than 37 of 158 eligible voters voted is $\square$ (Round to four decimal places as needed.)

Solution

Solution Steps

Step 1: Calculate the mean ($\mu$) and standard deviation ($\sigma$)

Given that $n = 158$, $p = 0.22$, and $k = 37$, we calculate the mean ($\mu$) and standard deviation ($\sigma$) of the binomial distribution. \[\mu = n \times p = 158 \times 0.22 = 34.76\] \[\sigma = \sqrt{n \times p \times (1 - p)} = \sqrt{158 \times 0.22 \times (1 - 0.22)} = 5.207\]

Step 2: Apply continuity correction and calculate z-scores

For fewer than $k = 37$ successes, we use $k-0.5 = 36.5$ as the upper limit. \[z_{upper} = \frac{k-0.5 - \mu}{\sigma} = \frac{36.5 - 34.76}{5.207} = 0.334\]

Step 3: Find the probability

The probability of fewer than $k = 37$ successes is the area to the left of $z_{upper}$.