Confidence intervals using the t distribution. Chapter 6 t scores as estimates of z scores; t curves as approximations of z curves Estimated standard.

Confidence intervals using the t distribution

Chapter 6

t scores as estimates of z scores;

t curves as approximations of z curves

Estimated standard errors

CIs using the t distribution

Testing the no effect (null) hypothesis)

t scores indicate the distance and direction of a score from the sample

mean.• t scores are computed like Z scores.

sigma

muXZ

s

XXt

• Simply substitute X for mu and s for sigma. That is, substitute estimates for parameters.

More t scores

• As long as the raw scores are obtained from a random sample, then t scores are least squares, unbiased, consistent estimators of what the Z scores would be if we had all the scores in the population.

Any score can be translated to a t scores as long as you can estimate mu and sigma with X-bar and s.

Body and tails of the curve• The body of a curve is the area enclosed in a

symmetrical interval around the mean.

• The tails of a curve are the two regions of the curve outside of the body.

• The critical values of the t curves are the number of estimated standard deviations one must go from the mean to reach the point where 95% or 99% of the curve is in the body and 5% or 1% is in the two tails combined.

• The critical values of the t curve change depending on how many df there are for MSW and s.

We can define a curve by stating its critical values

• The Z curve can be defined as one of the family of t curves in which 95.00% of the curve falls within 1.960 standard deviations from the mean and 99.00% falls within 2.576 standard deviations from the mean.

• We can define t curves in terms of how many estimated standard deviations you must go from the mean before the body of the curve contains 95% and 99% of the curve and the combined upper and lower tails contain 5% and 1% respectively.

t curves

• t curves are used instead of the Z curve when you are using samples to estimate sigma2.

• Since we are estimating sigma instead of knowing it, t curves are based on less information than the z curve.

• Therefore, t curves partake somewhat of the rectangular (“I know nothing”) distribution and tend to be flatter than the Z curve.

• The more degrees of freedom for MSW, the better our estimate of sigma2.

• The better our estimate, the more t curves resemble Z curves.

1 df

To get 95% of the population in the body of the curve when there is 1 df of freedom, you go out over 12 standard deviations.

5 df

To get 95% of the population in the body of the curve when there are 5 df of freedom, you go out over 3 standard deviations.

t curves and degrees of freedomFrequency

score3 2 1 0 1 2 3

Standarddeviations

Critical values of the t curves• The following table defines t curves with 1 through 10,000 degrees of freedom• Each curve is defined by how many estimated

standard deviations you must go from the mean to define a symmetrical interval that contains a proportions of .9500 and .9900 of the curve, leaving proportions of .0500 and .0100 in the two tails of the curve (combined).

• Values for .9500/.0500 are shown in plain print. Values for .9900/.0900 and the degrees of freedom for each curve are shown in bold print.

df 1 2 3 4 5 6 7 8.05 12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306.01 63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355

df 9 10 11 12 13 14 15 16.05 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120.01 3.250 3.169 3.106 3.055 3.012 2.997 2.947 2.921

df 17 18 19 20 21 22 23 24.05 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064.01 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797

df 25 26 27 28 29 30 40 60.05 2.060 2.056 2.052 2.048 2.045 2.042 2.021 2.000.01 2.787 2.779 2.771 2.763 2.756 2.750 2.704 2.660

df 100 200 500 1000 2000 10000.05 1.984 1.972 1.965 1.962 1.961 1.960.01 2.626 2.601 2.586 2.581 2.578 2.576

Using the t tableYou can answer things like:• If we have 13 degrees of freedom then how far do we have to

go above the mean in order to have only 5% of the curve left in the tails? Out to a t score of 2.160

• How many estimated standard deviations do we have to go out in order to leave 1% of the scores in the tails with 3 degrees of freedom? 5.841

• With 10 degrees of freedom and a critical value of .05, is a t score of -2.222 inside the body or the tail of the t curve? How about with 11 df. With 10 df, inside the body. With 11 df outside the body in the tail.

• What are the critical values of a t curve with 20 df? 2.086 at .95/.05 and 2.845 at .99/.01.

Estimated distance of sample means from mu: estimated standard errors of the mean

• We can compute the standard error of the mean when we know sigma.– We just have to divide sigma by the square root of n,

the size of the sample

• Similarly, we can estimate the standard error of the mean, the estimated average unsquared distance of sample means from mu.– We just have to divide s by the square root of n, the

size of the sample in which we are interested

Note that the estimated standard error is determined by only two factors: the estimated average unsquared distance of scores from mu

(s) and the size of the sample (n).

nssX /

  s n•  • A 2.83 8 1.00 = 2.83/2.83• B 12.00 8 4.24 = 12.00/2.83• C 20.00 8 7.07 = 20.00/2.83•  • D 2.83 1 2.83 = 2.83/1.00 • E 2.83 2 2.00 = 2.83/1.41• F 2.83 8 1.00 = 2.83/2.83• G 2.83 40 0.45 = 2.83/6.32

nssX /

Confidence intervals using the t distribution

There are two reasons to create confidence intervals with the t distribution

• 1. To test a theory about what mu is. (We call the theoretical population means muT).

• THAT’S THE IMPORTANT REASON AND WE WILL LEARN ABOUT IT FIRST

• 2. The other reason is to define an interval in which we are confident mu would fall if we knew it.

Confidence intervals around muT

Confidence intervals and hypothetical means

• We frequently have a theory about what the mean of a distribution should be.

• To be scientific, that theory about mu must be able to be proved wrong (falsified).

• One way to test a theory about a mean is to state a range where sample means should fall if the theory is correct.

• We usually state that range as a 95% confidence interval.

• To test our theory, we take a random sample from the appropriate population and see if the sample mean falls where the theory says it should, inside the confidence interval.

• If the sample mean falls outside the 95% confidence interval established by the theory, the evidence suggests that our theoretical population mean and the theory that led to its prediction is wrong.

• When that happens our theory has been falsified. We must discard it and look for an alternative explanation of our data.

For example:• For example, let’s say that we had a new

antidepressant drug we wanted to peddle. Before we can do that we must show that the drug is safe.

• Drugs like ours can cause problems with body temperature. People can get chills or fever.

• We want to show that body temperature is not effected by our new drug.

Testing a theory

• “Everyone” knows that normal body temperature for healthy adults is 98.6oF.

• Therefore, it would be nice if we could show that after taking our drug, healthy adults still had an average body temperature of 98.6oF.

• So we might test a sample of 16 healthy adults, first giving them a standard dose of our drug and, when enough time had passed, taking their temperature to see whether it was 98.6oF on the average.

Testing a theory - 2• Of course, even if we are right and our drug has no

effect on body temperature, we wouldn’t expect a sample mean to be precisely 98.600000…

• We would expect some sampling fluctuation around a population mean of 98.6oF.

• So, if our drug does not cause change in body temperature, the sample mean should be close to 98.6. It should, in fact, be within the 95% confidence interval around muT, 98.6.

• SO WE MUST CONSTRUCT A 95% CONFIDENCE INTERVAL AROUND 98.6o AND SEE WHETHER OUR SAMPLE MEAN FALLS INSIDE OR OUTSIDE THE CI.

To create a confidence interval around muT, we must estimate sigma from a sample.

• We randomly select a group of 16 healthy individuals from the population.

• We administer a standard clinical dose of our new drug for 3 days.

• We carefully measure body temperature.• RESULTS: We find that the average body

temperature in our sample is 99.5oF with an estimated standard deviation of 1.40o (s=1.40).

• IS 99.5oF. IN THE 95% CI AROUND MUT???

Knowing s and n we can easily compute the estimated standard error of the mean.

• Let’s say that s=1.40o and n = 16:

• = 1.40/4.00 = 0.35

• Using this estimated standard error we can construct a 95% confidence interval for the body temperature of a sample of 16 healthy adults.

nssX /

We learned how to create confidence intervals with the Z distribution in Chapter 4.

95% of sample means will fall in a symmetrical interval around mu that goes from 1.960 standard

errors below mu to 1.960 standard errors above mu

• A way to write that fact in statistical language is:

CI.95: mu + ZCRIT* sigmaX-bar or

CI.95: mu - ZCRIT* sigmaX-bar < X-bar < mu + ZCRIT* sigmaX-bar

For a 95% CI, ZCRIT = 1.960

But when we must estimate sigma with s, we must use the t distribution to define critical intervals around mu or muT.

Here is how we would write the formulae substituting t for Z and s for sigma

CI95: muT + tCRIT* sX-bar or

CI.95: muT - tCRIT* sX-bar < X-bar < muT + tCRIT* sX-bar

Notice that the critical value of t that includes 95% of the sample means changes with the number of degrees of freedom for s, our estimate of sigma, and must be taken from the t table.

If n= 16 in a single sample, dfW=n-k=15.

df 1 2 3 4 5 6 7 8.05 12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306.01 63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355

df 9 10 11 12 13 14 15 16

.05 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120

.01 3.250 3.169 3.106 3.055 3.012 2.997 2.947 2.921

df 17 18 19 20 21 22 23 24.05 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064.01 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797

df 25 26 27 28 29 30 40 60.05 2.060 2.056 2.052 2.048 2.045 2.042 2.021 2.000.01 2.787 2.779 2.771 2.763 2.756 2.750 2.704 2.660

df 100 200 500 1000 2000 10000.05 1.984 1.972 1.965 1.962 1.961 1.960.01 2.626 2.601 2.586 2.581 2.578 2.576

So, muT=98.6, tCRIT=2.131, s=1.40, n=16

Here is the confidence intervalCI.95: muT + tCRIT* sX-bar = = 98.6 + (2.131)*(1.40/ ) = = 98.6 + (2.131)*(1.40/4)

= 98.6 + (2.131)(0.35) = 98.60+ 0.75

CI.95: 97.85 < X-bar < 99.35

Our sample mean was 99.5o F. It falls outside the CI.95 and falsifies the theory that our drug has no effect on body temperature. Our drug may cause a slight fever.

16

Testing the no-effect (null) hypothesis• Specify what you believe the value of a statistic will be if

your idea of the situation is correct. In this case, you specified muT=98.6o F.

• Define a 95% Confidence Interval around that value of your test statistic.

• See if the test statistic falls inside the CI.95

• If it falls within the CI.95, you retain the no-effect hypothesis

• If it falls outside the CI.95, you reject the no effect hypothesis.Further, if you then have to guess what the true value of the test statistic is in the population as a whole, you chose the value that was found in the random sample.

You try one

• Instead of the results we just got, lets say we studied a sample of 9 randomly selected participants under the same conditions as in the example above and again found that the sample mean (X-bar) was 99.5oF and s=1.40. Compute the 95% confidence interval around 98.6?

• Can you falsify the hypothesis that your new drug has no effect on body temperature now?

df 1 2 3 4 5 6 7 8

.05 12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306

.01 63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355

df 9 10 11 12 13 14 15 16.05 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120.01 3.250 3.169 3.106 3.055 3.012 2.997 2.947 2.921

df 17 18 19 20 21 22 23 24.05 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064.01 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797

df 25 26 27 28 29 30 40 60.05 2.060 2.056 2.052 2.048 2.045 2.042 2.021 2.000.01 2.787 2.779 2.771 2.763 2.756 2.750 2.704 2.660

df 100 200 500 1000 2000 10000.05 1.984 1.972 1.965 1.962 1.961 1.960.01 2.626 2.601 2.586 2.581 2.578 2.576

muT=98.6, tCRIT=2.306, s=1.40, n=16

Here is the confidence intervalCI.95: muT + tCRIT* sX-bar = 98.6 + (2.306)*(1.40/ ) = 98.6 + (2.306)(1.40/3)

=98.6+ (2.306)(0.47) = 98.60+ 1.08

CI.95: 97.52 < X-bar < 99.68

This time, the mean of our smaller sample (99.5oF) falls inside the CI.95. These data are consistent with the hypothesis that our drug has no effect on body temperature.

9

Notice what happened• When decreased from 16 to 9, both the critical

value for t and the size of the standard error increased. The confidence interval therefore became wider (less precise) and we failed to falsify the “no effect” hypothesis.

• That’s fine for us in this case. But the FDA would probably not accept the smaller study because it is not sensitive enough (lacks sufficient power to reject a fale “no effect” hypothesis

• The general rule is”the bigger the n, the more sensitive the study.” That is, larger studies give you greater power to reject false hypotheses.

In fact, we usually want to falsify the “no effect” hypothesis

• Let’s say we were studying our drug’s ability to cause positive benefits (e.g., make depressed peoples’ moods better).

• We would want to be able to falsify the hypothesis of “no effect.”

• The “no effect” hypothesis is formally called “the null hypothesis.”

• So usually, having a more sensitive, more powerful study that forces the null hypothesis to be more precise, would be desirable.

• We’ll see the null hypothesis again and again from Chapter 8 on.

• THE FOLLOWING MATERIAL IS LESS IMPORTANT AND WILL ONLY BE GONE OVER IF THERE IS EXTRA TIME. MAKE SURE YOU UNDERSTAND TESTING THE NO EFFECT HYPOTHESIS BEFORE GOING ON.

Interval estimates for mu based on a random sample

Since Chapter 1 you have been making least squared, unbiased estimates

• You have learned to predict that everyone would score at a specific point. This is called “making point estimates.”

• The point estimates have been wrong in a least squared, consistent, unbiased way.

• If you want to be right, not wrong, you can’t make point estimates, you must make interval estimates.

Interval estimates of mu

• As you know, X-bar is our least squared, unbiased, consistent estimate of mu.

• To create an interval estimate of mu, we create a symmetrical interval around X-bar.

• We usually create 95% and/or 99% CIs to define that interval.

The generic formula for confidence intervals for mu

CI: X-bar + tCRIT* sX-bar or

CI: X-bar - tCRIT* sX-bar < mu< X-bar+ tCRIT* sX-bar

If we use the critical values of t at .95 we will have an interval that includes mu if our sample is one of the 95% that falls within a 95% CI around mu.

If we use the critical values of t at .99 we will have an interval that includes mu if our sample is one of the 95% that falls within a 99% CI around mu.

Here is an example

• Let’s say (because professors know everything) that I knew that the average height of all male Rutgers Juniors is 70.0 inches (5’10”)

• I assign you, the student to take a sample of 16 male Rutgers Juniors and measure their height.

• Then you must define an interval in which mu should be found.

• You find that X-bar = 69.3” and s = 2.00 inches.

df 1 2 3 4 5 6 7 8.05 12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306.01 63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355

df 9 10 11 12 13 14 15 16

.05 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120

.01 3.250 3.169 3.106 3.055 3.012 2.997 2.947 2.921

df 17 18 19 20 21 22 23 24.05 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064.01 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797

df 25 26 27 28 29 30 40 60.05 2.060 2.056 2.052 2.048 2.045 2.042 2.021 2.000.01 2.787 2.779 2.771 2.763 2.756 2.750 2.704 2.660

df 100 200 500 1000 2000 10000.05 1.984 1.972 1.965 1.962 1.961 1.960.01 2.626 2.601 2.586 2.581 2.578 2.576

So, X-bar=69.3, tCRIT=2.131, s=2.00, n=16

Here is the 95% confidence intervalCI.95: X-bar + tCRIT* sX-bar =

= 69.3 + (2.131)*(2.00/ ) =

= 69.3 + (2.131)*(2.00/4)

= 69.3 + (2.131)(0.50) = 69.30+ 1.16

CI.95: 68.14 < mu< 70.46

In this example, we knew mu ahead of time (70.0) and the computations were just an exercise. But exercise or not, our interval does include mu.

16

Here is a 99% CI with the same data X-bar=69.3, tCRIT=2.947, s=2.00, n=16

CI.99: X-bar + tCRIT* sX-bar =

= 69.3 + (2.947)*(2.00/4)

= 69.3 + (2.947)(0.50) = 69.30+ 1.47

CI.99: 67.83 < mu< 70.77

Of course, since we have to go further out into the tails of the curve to include 99% of the sample means, the interval for the CI99 is wider (less precise) that the interval for the CI.95 .

You try these

• Instead of the results we just got, lets say we studied a sample of 25 randomly selected male Rutgers Juniors and again found that the sample mean (X-bar) was 69.3 inches and s=2.00. Compute the 95% confidence and 99% confidence intervals for mu given these data.

• Would you expect the intervals to be wider or narrower that those you found for n=16?

df 1 2 3 4 5 6 7 8.05 12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306.01 63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355

df 9 10 11 12 13 14 15 16.05 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120.01 3.250 3.169 3.106 3.055 3.012 2.997 2.947 2.921

df 17 18 19 20 21 22 23 24

.05 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064

.01 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797

df 25 26 27 28 29 30 40 60.05 2.060 2.056 2.052 2.048 2.045 2.042 2.021 2.000.01 2.787 2.779 2.771 2.763 2.756 2.750 2.704 2.660

df 100 200 500 1000 2000 10000.05 1.984 1.972 1.965 1.962 1.961 1.960.01 2.626 2.601 2.586 2.581 2.578 2.576

So, X-bar=69.3, tCRIT=2.064, s=2.00, n=25

Here is the 95% confidence intervalCI.95: X-bar + tCRIT* sX-bar =

= 69.3 + (2.064)*(2.00/ ) =

= 69.3 + (2.064)*(2.00/5)

= 69.3 + (2.064)(0.40) = 69.30+ 0.83

CI.95: 68.47 < mu< 70.13

The interval is getting narrower isn’t it.

25

Here is a 99% CI with the same data X-bar=69.3, tCRIT=2.947, s=2.00, n=16

CI.99: X-bar + tCRIT* sX-bar =

= 69.3 + (2.797)*(2.00/5)

= 69.3 + (2.797)(0.40) = 69.30+ 1.12

CI.99: 68.18 < mu< 70.748

Notice that the CI .99 with n=25 is almost exactly as wide as the CI .95 with n=16.