Question

The amount of tea leaves in a can from a particular production line is normally distributed with μ = 110 grams and σ = 25 grams. A sample of 25 cans is to be selected. So, the middle 70% of all sample means will fall between what two values? A) 101.8 and 111.2 B) 104.8 and 115.2 C) 107.8 and 111.2 D) 101.8 and 119.2

Answer 1

B) 104.8 and 115.2

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.

In this problem, we have that:

`\mu = 110, \sigma = 25, n = 25, s = (25)/(√(25)) = 5`

So, the middle 70% of all sample means will fall between what two values?

50 - (70/2) = 15th percentile

50 + (70/2) = 85th percentile

15th percentile

X when Z has a pvalue of 0.15. So X when Z = -1.037.

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

By the Central Limit Theorem

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

`-1.037 = (X - 110)/(5)`

`X - 110 = -5*1.037`

`X = 104.8`

85th percentile

X when Z has a pvalue of 0.85. So X when Z = 1.037.

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

`1.037 = (X - 110)/(5)`

`X - 110 = 5*1.037`

`X = 115.2`

So the correct answer is:

B) 104.8 and 115.2

Answer 2

B)

Explanation:

The middle 70% of all sample means will fall between what two values (lower bound and higher bound)?

You only need to go to the Table of Z and find to 70% the value from Z. I attached this image.

So, the Z-values for middle 70% is equal to (-1.036, 1.036)

We can now make the upper limit and lower limit for the values. That is:

`\alpha = \mu-z*(/frac{sigma}{√(n))`

`\alpha_1 = 110 -1.036*5 =104.82`

`\alpha_2 = 110+1.036*5 =115.18`

Our interval is (104.8,115.2)