Inferring the Mean and Standard Deviation of a Population

Central Problem

Two important numbers tell us a lot about a distribution of data:

Mean tells us the central tendency of the data Standard deviation tells us the spread in the data

The problem is … we don’t normally know either of these and must infer them from a SRS of the population

Baby Paradox

Two hospitals in the same city deliver, on average, a 50:50 ratio of baby girls and baby boys. Hospital A delivers 120 babies a day (on average) while hospital B delivers 12 babies a day (on average). One day there were twice as many boys as girls born in one of the hospitals. In which hospital is this more likely to happen?

Measuring the mean…

How do we know the mean of a population?

Answer: We can either measure every single sample in the population or estimate the mean from a suitable SRSWe will assume that the population is normally

distributed so X has a normal distribution N(,/√n)

Standard Error and Standard Deviation These are two very distinct and different ideas:

Standard error measures the uncertainty in the measure of the mean

This depends on how YOU measure and sample size

Standard deviation measures the spread in the data This is a property of the data set – does not change

We can often estimate the standard deviation by measuring the standard error.

Standard error is always lessthan standard deviation

SE gets smaller as n grows

does not change!

SE measures the uncertaintyin location of mean

measures spread in data

t-Distributions

If we know then setting a confidence interval on how well our sample mean X measures the true mean is easy:

But – if we don’t know then we estimate use the t-distribution:

xz

n

Xt

sn

Closer look at t-distributions

The t-distribution looks very much like the Normal distribution and as the number of degrees of freedom (df) gets large the two become indistinguishable

t-distribution tables are used much the same way as N(0,1) – major difference is the df value

Xt

sn

Example…

You are inspecting a shipment of 10 000 precision machined rods to be used in an engine assembly plant. You select a random sample of 20 and measure the diameters. You find that the average diameter of the sample is 5.465 cm with a standard deviation in the measurements of 0.005 cm. It is critical that the diameters do not exceed 5.471 cm. You are willing to accept a 1% failure rate. Should you accept the shipment?

Solution: This would be an example of a 1-tailed t-

distribution, = 0.01, t19,0.01= 2.539

A 1% failure rate looks like this:

Test the numbers…

This implies that 99.998% of the sample will not exceed the threshold diameter

Accept!

5.471

5.465

0.005

(5.471 5.465)5.231

0.00519

X cm

cm

s cm

t

Two-tailed t-Tests In the previous example we

looked at whether or not the diameter was less than a maximum allowable value. Just as we have done earlier with confidence intervals we can also specify a maximum allowable range (“plus or minus”) for our mean.

Let’s test the mean diameter at a 95% confidence level that is implied by our measurement

Use following formula:

1, / 2 1, / 2n n

s sx t x t

n n

Margin of error

We measured mean diameter as 5.645 cm, s = 0.005 so the upper and lower margins are:

We can be 95% confident that the diameters of the parts are in the range (5.463,5.467) cm

1, / 2

0.005(2.093) 0.0024

19n

st

n

Example 7.9

Plot data: Identify variables, etc:

df = (50-1) = 49 = 0.05 = 23.56, s = 12.52 t = 2.009

Interval = (20.00,27.12)

?

Xt

sn

Example of a Matched Pairs t-test: Exercise 7.40 Formulate appropriate

hypothesesH0: no difference

H: LH > RH

Re-arrange data: find and s (see next page)

Ho: = 0 df = 25 - 1 = 24 Find

Use Excel =tdist(t, df, #tails)Use Table D

The probability of the null hypothesis is only 0.004

LH thread takes longer

2.844X

tsn

Robustness…

A statistical test is considered robust if: It is insensitive to deviations from original

assumptions being made. This could include smaller sample size or deviation from normality

Rules of thumb – When to use the t-test

• Small sample sizes (n≈15) and close to normal

• Mid range sample size (n ≥ 15) as long as distribution not strongly skewed and no outliers

• Large sample size (n > 40) even if skewed or with some outliers

Fine print: Rules of thumb do not obviate the need to always inspect your data! Stemplots or histograms give you insight into just how “skewed” or “outlier-riddled” is your data. Always know what the data set looks like before applying tests.

In conclusion…

Read 7.1 carefully – we skipped over some terms and discussions of applicability of the t-test

Be sure you understand when (and why) we need the t-test

Know the difference between standard deviation and Standard Error

Try: 7.4, 7.12, 7.26, 7.42