AP STATISTICS LESSON 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 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.

Common Confidence Levels Confidence levels tail area z* 90% 0.05 1.645 95% 0.025 1.96 99% 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 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.

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 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

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