Men consume on average 15 grams of protein a day. Assume a normal distribution with a standard deviation of 3 grams. A sample of 40 men was studied. What is the probability that the sample mean is between 15 and 16 grams per day

Respuesta :

Using the normal distribution, there is a 0.4826 = 48.26% probability that the sample mean is between 15 and 16 grams per day.

Normal Probability Distribution

The z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

  • The z-score measures how many standard deviations the measure is above or below the mean.
  • Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.
  • By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For this problem, the parameters are given as follows:

[tex]\mu = 15, \sigma = 3, n = 40, s = \frac{3}{\sqrt{40}} = 0.4743[/tex]

The probability is the p-value of Z when X = 16 subtracted by the p-value of Z when X = 15, hence:

X = 16:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{16 - 15}{0.4743}[/tex]

Z = 2.11

Z = 2.11 has a p-value of 0.9826.

X = 15:

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{15 - 15}{0.4743}[/tex]

Z = 0

Z = 0 has a p-value of 0.5.

0.9826 - 0.5 = 0.4826 = 48.26% probability that the sample mean is between 15 and 16 grams per day.

More can be learned about the normal distribution at https://brainly.com/question/15181104

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