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Chapter 10. Section 10.1 Part 2 – Estimating With Confidence. Confidence Interval for a Population Mean.
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Chapter 10 Section 10.1 Part 2 – Estimating With Confidence
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 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 will be approximately normal if n is sufficiently large.
Conditions for Constructing a Confidence Interval for • The construction of a confidence interval for a population mean 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 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.
Example 10.4 - Finding z* • To find an 80% confidence interval, we must catch the central 80% of the normal sampling distribution of . • 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. z*
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 (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.
Example 10.6 -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 is σ/√ n , and its mean is the population mean . • So there is probability C that the observed sample mean takes a value between: – z*σ√ n and + σ/√n • Whenever this happens, the population mean is contained between – z*σ√ n and + z*σ/√n
Example 10.5 -Video Screen Tension • 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. 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 rests on the fact that the statistic used to estimate has a normal distribution. • Because many sample statistics have normal distributions (approximately), confidence intervals have the form: estimate ± z* σestimate
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 . It is easier to pin down 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 - 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. • Read example 10.6 on p.550
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 -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? • Read example 10.7 on p.551-552
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 (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.
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 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 skewnessand 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 ±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.
