Download presentation

Presentation is loading. Please wait.

1
**AP STATISTICS LESSON 10 – 1 (DAY 2)**

CONFIDENCE INTERVAL FOR POPULATION MEAN

2
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.

3
**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.

4
**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.

5
**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.

6
**Example 10.4 Page 544 Finding 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.

7
**Common Confidence Levels**

Confidence levels tail area z* 90% 95% 99% Notice that for 95% confidence we use z* = This is more exact than the approximate value z*= 2 given by the rule.

8
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.

9
**Figure 10.6 Page 545 Changing 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.

10
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.

11
**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

12
**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.

13
**Example 10.5 Page 546 Video 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. Normal Probability Plot Stem Plot

14
**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.

15
**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

Similar presentations

© 2022 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google