Question

The normal daily human potassium requirement is in the range of 2000 to 6000 milligrams (mg), with larger amounts required during hot summer weather. The amount of potassium in food varies, but bananas are often associated with high potassium, with approximately 422 mg in a medium-sized banana. Suppose the distribution of potassium in a banana is normally distributed, with mean equal to 422 mg and standard deviation equal to 13 mg per banana. You eat n=3 bananas per day, and T is the total number of milligrams of potassium you receive from them. a. Find the mean and standard deviation of T. b. Find the probability that your total daily intake of potassium from the three bananas will exceed 1300 mg. (HINT: Note that T is the sum of three random variables

Answer 1

a) The mean is 1266 and the standard deviation is 22.52.

b) 6.55% probability that your total daily intake of potassium from the three bananas will exceed 1300 mg.

Explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For sums, we have that the mean is
`\mu*n` and the standard deviation is
`s = \sigma√(n)`

In this problem, we have that:

`\mu = 422, \sigma = 13`

a. Find the mean and standard deviation of T.

`n = 3`

So

Mean 3*422 = 1266,

Standard deviation
`s = 13*√(3) = 22.52`

The mean is 1266 and the standard deviation is 22.52.

b. Find the probability that your total daily intake of potassium from the three bananas will exceed 1300 mg.

This is 1 subtracted by the pvalue of Z when X = 1300. So

`Z = (X - \mu)/(\sigma)`

By the Central Limit Theorem

`Z = (X - \mu)/(s)`

`Z = (1300 - 1266)/(22.52)`

`Z = 1.51`

`Z = 1.51` has a pvalue of 0.9345

1 - 0.9345 = 0.0655

6.55% probability that your total daily intake of potassium from the three bananas will exceed 1300 mg.