Transcript 10-1 Day 3
AP STATISTICS LESSON 10 – 1 ( DAY 3 ) CHOOSING THE SAMPLE SIZE ESSENTIAL QUESTION: How are sample sizes determined? Objectives: • To find appropriate sample sizes mathematically. • To become aware of cautions involved in creating confidence intervals. How Confidence Intervals Behave 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 Making the Margin of Error Smaller 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. Making Margin of Error Smaller (continued…) To obtain a smaller margin of error from the same data you must be willing to accept lower confidence: • σ gets smaller. The standard deviation σ measures the variation in the population. You can think of the variation among individuals in the population as noise that obscures the average μ. It is easier to pin down μ when σ is small. • n gets larger. Increasing the sample size n reduces the margin of error for any fixed confidence level. Because n appears under a square root sign, we must take four times as many observations in order to cut the margin of error in half. Example 10.6 Page 550 Changing the Confidence Level 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 10.5. Choosing the Sample Size 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. Example 10.7 Page 551 Determining Sample Size 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? Sample Size for Desired Margin of Error 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 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. Some Cautions • Any formula for inference is correct only in specific circumstances. • The data must be an SRS from the population. • The formula is not correct for probability sampling designs more complex than an SRS. • There is no correct method for inference from data haphazardly collected with bias of unknown size. • Because x is strongly influenced by a few extreme observations, outliers can have a large effect on the confidence interval. You should search for outliers and try to correct them or justify their removal before computing the interval. Cautions (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 shewness 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. Cautions (continued…) • You must know the standard deviation σ of the population. This unrealistic requirement renders the interval x ±σ/√n of little use in statistical practice. • The margin of error in a confidence interval covers only random sampling errors. • Practical difficulties, such as undercoverage and nonresponse in a sample survey, can cause additional errors that may be larger than random sampling error.