normal distribution statistics question

Question

A manufacturer knows that their items have a normally distributed length, with a mean of 14.8 inches, and standard deviation of 0.9 inches.

If 16 items are chosen at random, what is the probability that their mean length is less than 14.2 inches?

Answer 1

Because we have a normal distribution we can use the normal distribution probability table to compute the probability.

First we need to compute the z-statistic corresponding to our sample:

z = sample mean - population mean

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population standard deviation/sqrt(sample size)

= 14.2 - 14.8 = -2.67

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0.9/sqrt(16)

since we want to find the probability the sample mean is less than 14.2, we need to compute

P(z < -2.67) which from the normal probability tables is 0.004.

Answer 2

Given:

n=16

μ=14.8

σ=0.9

Want: P( Xbar < 14.2)

Xbar ~ Normal (Mean=μ = 14.8, SD= σ/√ n = 0.9/√ 16 = 0.225)

One way to calculate the probability via TI-83 or 84

P( Xbar < 14.2)=normalcdf(-1E99, 14.2, 14.8,0.225) ~ 0.004

Or use software/online websites like https://stattrek.com/online-calculator/normal.aspx