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sukhopar [10]
1 year ago
12

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
Mathematics
1 answer:
tatyana61 [14]1 year ago
4 0

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

<h3>Normal Probability Distribution</h3>

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

Z = \frac{X - \mu}{\sigma}

  • 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 s = \frac{\sigma}{\sqrt{n}}.

For this problem, the parameters are given as follows:

\mu = 15, \sigma = 3, n = 40, s = \frac{3}{\sqrt{40}} = 0.4743

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

X = 16:

Z = \frac{X - \mu}{\sigma}

By the Central Limit Theorem

Z = \frac{X - \mu}{s}

Z = \frac{16 - 15}{0.4743}

Z = 2.11

Z = 2.11 has a p-value of 0.9826.

X = 15:

Z = \frac{X - \mu}{s}

Z = \frac{15 - 15}{0.4743}

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 brainly.com/question/15181104

#SPJ1

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