Questions: Assume that the amounts of weight that male college students gain during their freshman year are normally distributed with a mean of μ=1.3 kg and a standard deviation of σ=4.4 kg. Complete parts (a) through (c) below. a. If 1 male college student is randomly selected, find the probability that he gains between 0 kg and 3 kg during freshman year. The probability is 0.2666 . (Round to four decimal places as needed.) b. If 16 male college students are randomly selected, find the probability that their mean weight gain during freshman year is between 0 kg and 3 kg . The probability is (Round to four decimal places as needed.)

Assume that the amounts of weight that male college students gain during their freshman year are normally distributed with a mean of μ=1.3 kg and a standard deviation of σ=4.4 kg. Complete parts (a) through (c) below.
a. If 1 male college student is randomly selected, find the probability that he gains between 0 kg and 3 kg during freshman year.

The probability is 0.2666 .
(Round to four decimal places as needed.)
b. If 16 male college students are randomly selected, find the probability that their mean weight gain during freshman year is between 0 kg and 3 kg .

The probability is
(Round to four decimal places as needed.)

Transcript text: Assume that the amounts of weight that male college students gain during their freshman year are normally distributed with a mean of $\mu=1.3 \mathrm{~kg}$ and a standard deviation of $\sigma=4.4 \mathrm{~kg}$. Complete parts (a) through (c) below. a. If 1 male college student is randomly selected, find the probability that he gains between 0 kg and 3 kg during freshman year. The probability is 0.2666 . (Round to four decimal places as needed.) b. If 16 male college students are randomly selected, find the probability that their mean weight gain during freshman year is between 0 kg and 3 kg . The probability is $\square$ (Round to four decimal places as needed.)

Solution

Solution Steps

Step 1: Convert the range values to Z-scores

Using the formula $Z = \frac{X - \mu}{\sigma}$, convert the range values to Z-scores. For the lower bound 0, $Z_1 = \frac{0 - 1.3}{4.4} = -0.295$. For the upper bound 3, $Z_2 = \frac{3 - 1.3}{4.4} = 0.386$.

Step 2: Find the probability corresponding to these Z-scores

Using Z-tables or software, find the probability for $Z_1$ and $Z_2$. The probability that an observation falls within the specified range is $P(Z_1 < Z < Z_2) = P(Z < Z_2) - P(Z < Z_1) = 0.267$.

Final Answer:

The probability that the observation or mean of a sample falls within the specified range is approximately 0.267.

Step 1: Apply the Central Limit Theorem (CLT)

Given $n = 16$, the distribution of sample means is approximately normal with mean $\mu_{\bar{x}} = 1.3$ and standard deviation $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = 1.1$.

Step 2: Convert the range values to Z-scores for the sample mean

Using the formula $Z = \frac{\bar{X} - \mu_{\bar{x}}}{\sigma_{\bar{x}}}$, convert the range values to Z-scores. For the lower bound 0, $Z_1 = \frac{0 - 1.3}{1.1} = -1.182$. For the upper bound 3, $Z_2 = \frac{3 - 1.3}{1.1} = 1.546$.

Step 3: Find the probability corresponding to these Z-scores

Using Z-tables or software, find the probability for $Z_1$ and $Z_2$. The probability that the mean of a sample falls within the specified range is $P(Z_1 < Z < Z_2) = P(Z < Z_2) - P(Z < Z_1) = 0.82$.

Final Answer:

The probability that the observation or mean of a sample falls within the specified range is approximately 0.82.

Was this solution helpful?