AP STATISTICSLESSON 10 – 1

(DAY 2)

CONFIDENCE INTERVAL FOR POPULATION MEAN

ESSENTIAL QUESTION:

What are the formulas used and conditions necessary to construct confidence intervals?

Objectives:

• To construct confidence intervals.

• To recognize conditions in which confidence intervals can be used.

Confidence Interval for a Population Mean

When a sample of size n comes from a SRS, the construction of the confidence interval depends on the fact that the sampling distribution of the sample mean x is at least approximately normal.

This distribution is exactly normal if the population is normal.

When the population is not normal, the central limit theorem tells us that the sampling distribution of x will be approximately normal if n is sufficiently large.

Conditions for Constructing a Confidence Interval for μ

• The construction of a confidence interval for a population mean μ is approximate when:

• The data come from an SRS from the population of interest, and

• The sampling distribution of x is approximately normal.

Confidence Interval Building Strategy

Our construction of a 95% confidence interval for the mean SAT Math score began by noting that any normal distribution has probability about 0.95 within 2 standard deviations of its mean.

To do that , we must go out z* standard deviations on either side of the mean.

Since any normal distribution can be standardized, we can get the value z* from the standard normal table.

Example 10.4 Page 544Finding z*

To find an 80% confidence interval, we must catch the central 80% of the normal sampling distribution of x.

In catching the central 80% we leave out 20%, or 10% in each tail.

So z* is the point with 10% area to its right.

Common Confidence Levels

Confidence levels tail area z*

90% 0.05 1.64595% 0.025 1.9699% 0.005 2.576

Notice that for 95% confidence we use z* = 1.960. This is more exact than the

approximate value z*= 2 given by the 68-95- 99.7 rule.

Table C

The bottom row of the C table can be used to find some values of z*. Values of z* that mark off a specified area under the standard normal curve are often called critical values of the distribution.

Figure 10.6 Page 545Changing the Confidence Level

In general, the central probability C under a standard normal curve lies between –z* and z*.

Because z* has area (1-C)/2 to its right under the curve, we call it the upper (1-C)/2 critical value.

Critical Value The number z* with probability p lying to

its right under the standard normal curve is called the upper p critical value of the standard normal distribution.

Level C Confidence Intervals

• Any normal curve has probability C between the points z* standard deviations below its mean and the point z* standard deviations above its mean.

• The standard deviation of the sampling distribution of x is σ/√ n , and its mean is the population mean μ. So there is probability C that the observed sample mean x takes a value between μ – z*σ√ n and μ + σ/√ n

• Whenever this happens, the population mean μ is contained between x – z*σ√ n and x + z*σ/√n

Confidence Interval for a Population Mean

Choose an SRS of size n from a population having unknown mean μ and known standard deviation of σ. A level C confidence interval for μ is

x ± z*σ/√ n Here z* is the value with C between –z*

and z* under the standard normal curve. This interval is exact when the population distribution is normal and is approximately correct for large n in other cases.

Example 10.5 Page 546Video Screen Tension

• Step 1 – Identify the population of interest and the parameter you want to draw conclusions about.

• Step 2 – Choose the appropriate inference procedure. Verify the conditions for using the selected procedure.

• Step 3 – If conditions are met, carry out the inference procedure.

• Step 4 – Interpret your results in the context of the problem.

Stem Plot

Normal Probability Plot

Inference Toolbox:Confidence Intervals

To construct a confidence interval:

Step 1: Identify the population of interest and the parameters you want to draw conclusions about.

Step 2: Choose the appropriate inference procedure.

Verify the conditions for the selected procedure.

Step 3: if the conditions are met, carry out the inference procedure.

CI = estimate ± margin of error

Step 4: Interpret your results in the context of the problem.

Confidence Interval Form

The form of confidence intervals for the population mean μ rests on the fact that the statistic x used to estimate μ has a normal distribution.

Because many sample statistics have normal distributions (approximately), confidence intervals have the form:

estimate ± z* σ estimate