C HAPTER 10 Section 10.1 Part 2 – Estimating With Confidence

C ONFIDENCE I NTERVAL FOR A P OPULATION M EAN

C ONFIDENCE I NTERVAL B UILDING S TRATEGY Our construction of a 95% confidence interval for the mean Mystery Mean 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.

E XAMPLE F INDING Z * z*z*

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

T ABLE C The bottom row of the C table (back cover of book) 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.

E XAMPLE C HANGING THE C ONFIDENCE L EVEL 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.

C RITICAL V ALUE 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.

L EVEL C C ONFIDENCE I NTERVALS

E XAMPLE V IDEO S CREEN T ENSION Read example 10.5 on p.546: 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

I NFERENCE T OOLBOX : C ONFIDENCE I NTERVALS 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.

C ONFIDENCE I NTERVAL F ORM

H OW C ONFIDENCE I NTERVALS B EHAVE Properties that are shared by all confidence intervals: The user chooses the confidence level and the margin of error.. A small margin of error says that we have pinned down the parameter quite precisely. Margin of error = z*σ/√n

M AKING THE M ARGIN OF E RROR S MALLER z* and σ in the numerator and √n in the denominator will make the margin of error smaller when: z* gets smaller. Smaller z* is the same as smaller confidence level C. There is a trade-off between the confidence level and the margin of error.

M AKING M ARGIN OF E RROR S MALLER ( CONTINUED …)

E XAMPLE C HANGING THE C ONFIDENCE L EVEL Suppose that the manufacturer in example 10.5 wants a 99% confidence level. Find the new margin of error, confidence level and compare this level to the 90% confidence level found in Example Read example 10.6 on p.550

C HOOSING THE S AMPLE S IZE A wise user of statistics never plans data collection without planning the inference at the same time. You can arrange to have both high confidence and a small margin of error by taking enough observations.

E XAMPLE D ETERMINING S AMPLE S IZE Company management wants a report of the mean screen tension for the day’s production accurate to within ± 5 mV with a 95% confidence. How large a sample of video monitors must be measured to comply with this request? Read example 10.7 on p

S AMPLE S IZE FOR D ESIRED M ARGIN OF E RROR To determine the sample size n that will yield a confidence interval for a population mean with a specified margin of error m, set the expression for the margin of error ( m ) to be less than or equal to m and solve for n : z*σ/√n ≤ m In practice, taking observations costs time and money. Do notice once again that it is the size of the sample that determines the margin of error. The size of the population does not determine the size of the sample we need.

S OME C AUTIONS

C AUTIONS ( CONTINUED …) If the sample size is small and the population is not normal, the true confidence level will be different from the value C used in computing the interval. Examine your data carefully for skewness and other signs of nonnormality. When n ≥15, the confidence level is not greatly disturbed by nonnormal populations unless extreme outliers or quite strong skewness are present.

C AUTIONS ( CONTINUED …)

Homework: p.542- #’s 4, 6, 10, 13, 18, 22 & 24