To find the probability that the mean weight gain of 16 male college students is between 0 kg and 3 kg, we need to use the Central Limit Theorem to approximate the distribution of sample means.
The mean weight gain of 16 male college students will follow a normal distribution with mean μ = 1.3 kg and standard deviation σ/√n, where n is the sample size. In this case, n = 16 and σ = 4.4 kg.
To find the probability, we need to standardize the values using the standard normal distribution.
First, we calculate the standard deviation of the sample mean:
Standard deviation of the sample mean = σ/√n = 4.4/√16 = 1.1 kg
Next, we standardize the values:
Z1 = (0 - μ) / (σ/√n) = (0 - 1.3) / 1.1 = -1.18
Z2 = (3 - μ) / (σ/√n) = (3 - 1.3) / 1.1 = 1.55
Using a standard normal distribution table or a calculator, we can find the probabilities associated with these z-scores:
P(0 kg ≤ X ≤ 3 kg) = P(-1.18 ≤ Z ≤ 1.55)
Consulting the standard normal distribution table, we find that P(Z ≤ -1.18) = 0.1181 and P(Z ≤ 1.55) = 0.9394.
To find the difference between these probabilities, we subtract:
P(-1.18 ≤ Z ≤ 1.55) = P(Z ≤ 1.55) - P(Z ≤ -1.18) = 0.9394 - 0.1181 = 0.8213
Therefore, the probability that the mean weight gain of the 16 male college students is between 0 kg and 3 kg is 0.8213 (rounded to four decimal places).