OA Cancelled due to technical difficulties…

Tonight’s HW is A2: 541(11b = PANIC and 11d)We will start using the books next Friday!

Chapter 23: Confidence Intervals for a Population Mean In addition to using interval estimates for population proportions (p), we can also use confidence intervals to estimate the true value of a population mean (µ).

The formula for a CI for a population mean is based on the sampling distribution of the sample mean . Recall that:

is approximately normally distributed when n > 30 or when the population is normal.

Thus far, we have been using z-critical values from the normal distribution to make our confidence intervals.

Now, remember that a z-score is calculated as

As long as is normally distributed, the distribution of z will also be normal. Why?

Since , m s, n are all constants, the shape shouldn’t change, only the center and spread.

observed value-expected valuez =

standard deviation

y

n

However, if we don’t know s (the population standard deviation) and we use s (the sample standard deviation) to

estimate s, the distribution of isn’t quite normal, even if is approximately normal.

So, we give this quantity a new name:

t = ysn

If you have your book open it to the back and look at it now.

You will get t-tables on the AP exam and can also find them in the back of your book.

Since t is based on 2 variables (s and ) instead of just (like z),

the t-distributions have more variability than the z (normal) distribution.

However, as the sample size increases, s gets closer to s and the t-distributions get closer to the standard normal distribution (z).

 The t-curves are very similar to normal curves, except that they are wider and defined by a number called the DEGREES OF FREEDOM. For this chapter, df = n – 1.

Properties of the t-distributions:The t-curve corresponding to any

fixed number of degrees of freedom (df) is bell-shaped, symetric and centered at 0.

Each t-curve is more spread out than the z-curve (standard normal curve).

As the df increase, the spread of the corresponding t-curve decreases.

As the number of df increases, the t-curves get closer to the z-curve.

T-curves are wider than the z-curve, so we must go further than 1.96 SD to capture 95% of the area.

To find out how far, we use a t-critical value from the t-table.

If df = 20 and you want 95% confidence, what t-critical value should you use?

Thus, to capture the middle 95% of the t-distribution with 20 df, you must go out 2.086 stdev.

2.086

Find the t critical values for the following:a. 95% confidence with n = 10b. 90% confidence with n = 25  c. 99% confidence with n = 100

When using the t-table and the df you want are not provided, round down to the nearest df given.

2.262

1.7112.369

Suppose that a machine is designed to produce bolts that have a diameter of 5 mm. Every hour a random sample of 15 bolts is selected and a 95% confidence interval for the mean diameter is constructed. If there is evidence that m ≠ 5, the machine is adjusted. In one particular sample, the mean diameter was 5.08 mm with a standard deviation of 0.11 mm. Calculate the interval and decide if you need to adjust the machine.

It is asking for a confidence interval, what should we do?

That’s right, PANIC!

P: We want to estimate the true mean bolt diameters that the machine is producing (mD).

A: We have a random sample of 15

bolts.With a SRS, each bolt diameter is

independent of the other. Assuming there are at least 150

bolts produced each hour, our sample is less than 10% of the population.

LEN - …

3 ways to satisfy LEN:1) Data comes from a normal

population2) Graphed data is unimodal and

symmetric3) n > 30

None of these are true but…LEN: It is reasonable to assume that bolt diameters produced by the machine are normally distributed.

N: The conditions for a 95% 1-sample t interval have been met. DF = 14.

I:

=5.08 2.145(.11/√15)

= (5.091, 5.1409)C: I am 95% confident that the interval from 5.091 to 5.1409 captures the true mean diameter of bolts. Does the machine need to be adjusted?

Tonight’s HW: A2: 541(11b , d)

& stay safe…