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- 1. INTRODUCTION TO STATISTICS & PROBABILITY Chapter 6: Introduction to Inference (Part 1) Dr. Nahid Sultana 1
- 2. Chapter 6 Introduction to Inference 6.1 Estimating with Confidence 6.2 Tests of Significance 6.3 Use and Abuse of Tests 6.4 Power and Inference as a Decision 2
- 3. 3 6.1 Estimating with Confidence Inference Statistical Confidence Confidence Intervals Confidence Interval for a Population Mean Choosing the Sample Size
- 4. 4 Overview of Inference Methods for drawing conclusions about a population from sample data are called statistical inference Methods: Confidence Intervals - for estimating a value of a population parameter Tests of significance – which assess the evidence for a claim about a population Both are based on sampling distribution Both use probabilities based on what happen if we used the inference procedure many times.
- 5. 5 How Statistical Inference Works
- 6. 6 Statistical Estimation Estimating µ with confidence. Problem: population with unknown mean, µ Solution: Estimate µ with x But does not exactly equal to µx How accurately does estimate µ?x
- 7. 7 Since the sample mean is 240.79, we could guess that µ is “somewhere” around 240.79. How close to 240.79 is µ likely to be? To answer this question, we must ask: ?populationthefrom16sizeof SRSsmanytookweifvarymeansampletheHow would x Statistical Estimation .16sizeofSRSafor79240meansampleand ;20:ondistributipopulationtheSuppose n.x )N(µ, σ == =
- 8. 8 Statistical Estimation (Cont…) 10.and10-betweenbewillxall of95%rule,99.7–95–68Using + ◊ µµ .xofpoints10 withinisthatsayingassametheis of10withinliesxsay thatTo µ µ◊ 250.79.10xand230.7910-xbetweenliesmeanunknown that theconfident95%areesay that wWe240.79.xHere =+= = µ 5).,N(:xofonDistributi µ .of10withinis240.79x thatconfidence95%areweHere µ= ◊
- 9. 9 Confidence Interval estimate ± margin of error The sampling distribution of tells us how close to µ the sample mean is likely to be. All confidence intervals we construct will have the form: xx The estimate ( in this case) is our guess for the value of the unknown parameter. The margin of error (10 here) reflects how accurate we believe our guess is, based on the variability of the estimate, and how confident we are that the procedure will catch the true population mean μ. We can choose the confidence level C, but 95% is the standard for most situations. Occasionally, 90% or 99% is used. We write a 95% confidence level by C = 0.95. The interval of numbers between the values ± 10 is called a 95% confidence interval for μ. )10.xand10-xbetweenliesmeanthatconfident(95% +µ
- 10. 10 Confidence Level The sample mean will vary from sample to sample, but when we use the method estimate ± margin of error to get an interval based on each sample, C% of these intervals capture the unknown population mean µ. The 95% confidence intervals from 25 SRSs In a very large number of samples, 95% of the confidence intervals would contain μ.
- 11. 11 Confidence Interval for a Population Mean We will now construct a level C confidence interval for the mean μ of a population when the data are an SRS of size n. The construction is based on the sampling distribution of the sample mean .x This sampling distribution is exactly when the population distribution is N(µ,σ). By the central limit theorem, this sampling distribution is appt. for large samples whenever the population mean and s.d. are μ and σ. )σN(µ, n/ )σN(µ, n/ Normal curve has probability C between the point z∗ s.d. below the mean and the point z∗ s.d. above the mean. Normal distribution has probability about 0.95 within ±2 s.d. of its mean.
- 12. 12 Confidence Interval for a Population Mean (Cont…) 12 Values of z∗ for many choices of C shown at the bottom of Table D: Choose an SRS of size n from a population having unknown mean µ and known standard deviation σ. A level C confidence interval for µ is: The margin of error for a level C confidence interval for μ is n zx σ *± n zm σ *=
- 13. 13 Confidence Interval for a Population Mean (Cont…) )59.250,99.230(8.979.240 16 20 96.179.240* =±= ⋅±=⋅± n zx σ 79240meanSample .16sizeofSRS 20:ondistributiPopulation .x n );N(µ, σ = = = Calculate a 95% confidence interval for µ. n zx σ *±
- 14. 14 Confidence Interval for a Population Mean (Cont…) Margin of error for the 95% CI for μ: 19803.198 1200 3500 )960.1(* ≈=== n zm σ 95% CI for μ: )3371,2975(1983173 =±=± mx Example:. Let’s assume that the sample mean of the credit card debt is $3173 and the standard deviation is $3500. But suppose that the sample size is only 300. Compute a 95% confidence interval for µ. Margin of error for the 95% CI for μ: 396 300 3500 )960.1(* === n zm σ 95% CI for μ: )3569,2777(3963173 =±=± mx Example: A random pool of 1200 loan applicants, attending universities, had their credit card data pulled for analysis. The sample of applicants carried an average credit card balance of $3173. The s.d. for the population of credit card debts is $3500. Compute a 95% confidence interval for the true mean credit card balance among all undergraduate loan applicants.
- 15. 15 The Margin of Error How sample size affects the confidence interval. Sample size, n=1200; Margin of error, m= 198 Sample size, n=300; Margin of error, m= 396 n=300 is exactly one-fourth of n=1200. Here we double the margin of error when we reduce the sample size to one-fourth of the original value. A sample size 4 times as large results in a CI that is half as wide. CI for µ
- 16. 16 How Confidence Intervals Behave The confidence level C determines the value of z*. The margin of error also depends on z*. m = z *σ n C z*−z* m m The user chooses C, and the margin of error follows from this choice. We would like high confidence and a small margin of error. To reduce the margin of error: Use a lower level of confidence (smaller C, i.e. smaller z*). Increase the sample size (larger n). Reduce σ. High confidence says that our method almost always gives correct answers. A small margin of error says that we have pinned down the parameter quite precisely
- 17. 17 How Confidence Intervals Behave Example: Let’s assume that the sample mean of the credit card debt is $3173 and the standard deviation is $3500. Suppose that the sample size is only 1200. Compute a 95% confidence interval for µ. Margin of error for the 95% CI for μ: 198 1200 3500 )960.1(* === n zm σ 95% CI for μ: )3371,2975(1983173 =±=± mx Example: Compute a 99% confidence interval for µ. Margin of error for the 99% CI for μ: 260 1200 3500 )576.2(* === n zm σ 99% CI for μ: )3433,2913(2603173 =±=± mx The larger the value of C, the wider the interval.
- 18. 18 Impact of sample size The spread in the sampling distribution of the mean is a function of the number of individuals per sample. The larger the sample size, the smaller the s.d. (spread) of the sample mean distribution. The spread decreases at a rate equal to √n. Sample size n Standarddeviationσ⁄√n
- 19. 19 To obtain a desired margin of error m, plug in the value of σ and the value of z* for your desired confidence level, and solve for the sample size n. 2 * * =⇔= m z n n zm σσ * n zm σ = Example: Suppose that we are planning a credit card use survey as before. If we want the margin of error to be $150 with 95% confidence, what sample size n do we need? For 95% confidence, z* = 1.960. Suppose σ = $3500. 209254.2091 150 3500*96.1* 22 ≈= = = m z n σ Would we need a much larger sample size to obtain a margin of error of $100? Choosing the Sample Size

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