The population standard deviation is 8. A random sample of 64 has… Step by Step

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The question

Question: a sample of 64 has mean 50, and the SD of all the numbers is 8: find the 95% confidence interval for the true mean.

95% interval for μ: x̄ = 50, σ = 8, n = 64

x̄, the sample mean, is 50: the middle of our interval.

σ, the SD of all the numbers, is known: 8.

And n, the sample size, is 64.

x̄: the sample mean, the middle

σ: the SD of all the numbers (known)

n: the sample size

We do not know the true mean of all the numbers.

From one sample, we give a range that very likely holds it: a confidence interval.

It goes from the sample mean minus the error bound, EBM, to the sample mean plus the error bound.

interval = x̄ − EBM to x̄ + EBM

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Find z*

95 percent of the bell is in the middle.

That leaves 5 percent: 2.5 percent in each end.

95% in the middle: 5% left over, 2.5% in each end

z* marks the cut between the middle and the ends.

Left of z* are the middle and one end: 0.95 plus 0.025 is 0.9750.

left of z* = 0.95 + 0.025 = 0.9750

Now we look for 0.9750 inside the table.

0.9750 is in the table: row 1.9, column .06.

So z = 1.96.

z* = 1.96

2 Error bound

The error bound: z* times σ over the square root of n.

σ over the square root of n is the SD of sample means, from the sample means lesson.

z* of those SDs on each side holds the middle part we want.

EBM = z* × σ√n

The square root of 64 is 8.

8/8 is 1.

Times 1.96: 1.96.

EBM = 1.96 × 8√64 = 1.96 × 88 = 1.96

Careful!

We divide by the square root of 64, not by 64.

Careful: not 864 ✗

3 The interval

Take the error bound away from the middle: 50 minus 1.96 is 48.04.

x̄ − EBM = 50 − 1.96 = 48.04

And add it: 50 plus 1.96 is 51.96.

x̄ + EBM = 50 + 1.96 = 51.96

The 95 percent confidence interval is (48.04, 51.96).

(48.04, 51.96)

4 Say it in words

In words: we are 95 percent confident that the true mean is between 48.04 and 51.96.

95% confident: 48.04 < μ < 51.96

5 Check

Let's check: the middle of the interval should be 50.

48.04 plus 51.96, divided by 2, is 50.

middle: (48.04 + 51.96) ÷ 2 = 50

Half the width should be the error bound: 51.96 minus 48.04, divided by 2, is 1.96.

So it is right!

half the width: (51.96 − 48.04) ÷ 2 = 1.96

6 Pick the choice

Our answer is (48.04, 51.96).

That is choice A.

A) (48.04, 51.96)

7 Why not the others

B is (48.355, 51.645).

Wrong: used 1.645: that is for 90%.

B) (48.355, 51.645) ✗ used 1.645: that is for 90%

C is (34.32, 65.68).

Wrong: used the SD 8 without dividing by √64.

C) (34.32, 65.68) ✗ used the SD 8 without dividing by √64

D is (49, 51).

Wrong: used 1 SE.

D) (49, 51) ✗ used 1 SE

Let's look back.

First: find z*.

Then: error bound.

Then: the interval.

Then: say it in words.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer