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 SRS  We 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 less than standard deviation SE gets smaller as n grows  does not change! SE measures the uncertainty in 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:

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

Example… You are inspecting a shipment of 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 cm with a standard deviation in the measurements of cm. It is critical that the diameters do not exceed 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, t 19,0.01 = 2.539t 19,0.01 = A 1% failure rate looks like this:

Test the numbers… This implies that % of the sample will not exceed the threshold diameter Accept!

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: Margin of error

We measured mean diameter as cm, s = 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

Example 7.9 Plot data: Identify variables, etc:  df = (50-1) = 49   = 0.05   = 23.56, s =  t = Interval = (20.00,27.12) ?

Example of a Matched Pairs t-test: Exercise 7.40 Formulate appropriate hypotheses  H 0 : no difference  H  : LH > RH Re-arrange data:  find  and s (see next page)

H o :  = 0 df = = 24 Find  Use Excel =tdist(t, df, #tails)  Use Table D The probability of the null hypothesis is only LH thread takes longer

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