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Chapter 3 Basic Concepts in Statistics and Probability

Chapter 3 Basic Concepts in Statistics and Probability. 3.7 Choice of Statistical Distributions. The distributions are simply models of reality (not reality themselves) Always check the assumptions. 3.8 Statistical Inference.

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Chapter 3 Basic Concepts in Statistics and Probability

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  1. Chapter 3Basic Concepts in Statistics and Probability

  2. 3.7 Choice of Statistical Distributions • The distributions are simply models of reality (not reality themselves) • Always check the assumptions

  3. 3.8 Statistical Inference • Methods of Statistical Inference used to estimate the parameters of the statistical distributions: • Central Limit Theorem • Point Estimation • Maximum Likelihood Estimation • Confidence Intervals • Tolerance Intervals • Hypothesis Tests • Probability Plots • Likelihood Ratio Tests • Bonferroni Intervals

  4. 3.8.1 Central Limit Theorem • If X ~ N(, 2), then • If X ~ unknown, when n is large • If X distribution closes to normal, n ~ 15 – 20 • If X distribution differs from normal greatly, n ~ 100

  5. Central Limit Theorem The Central Limit Theorem Let X1,…,Xn be a sequence of independent random variables with means 12 … nand variances 12, 22 …n2,then Approaches the standard normal distribution as n approaches infinity.

  6. Rule of Thumb For most populations, if the sample size is greater than 30, the Central Limit Theorem approximation is good. Normal approximation to the Binomial: If X ~ Bin(n,p) and if np > 5, and n(1– p) > 5, then X ~ N(np, np(1-p)) approximately. Normal Approximation to the Poisson: If X ~ Poisson(λ), where λ > 10, then X ~ N(λ, λ2).

  7. 3.8.2 Point Estimation • The statistical distributions have 1 or more parameters (usually represented by Greek letters) • The value of these parameters are generally unknown and must be estimated by sample statistics. • Point estimator: the form of the estimation • Point estimates: the numerical value of an estimator

  8. 3.8.2.1 Maximum Likelihood Estimation • A method for obtaining point estimates such that the probability of observing the data that are in a sample is maximized. • Likelihood function: • Likelihood function for  of Normal distribution • Setting the derivative = 0, then

  9. 3.8.3 Confidence Intervals • An interval estimator will contain the unknown parameter with (approximately) a specified probability. • The desired degree of confidence determines the width of the interval. • For a fixed sample size, an increase in the width will increase the degree of confidence. • Increasing the sample size will decrease the width of a confidence interval. • Narrower confidence intervals are more meaningful (lower uncertainty) • However, a low level of uncertainty is contingent upon the requisite assumptions being met.

  10. Confidence Intervals • Confidence intervals that utilize either t or z are always of the form Where:  is the parameter to be estimated is the point estimator is the estimated standard deviation of the point estimator

  11. Constructing a CI To see how to construct a confidence interval, let  represent the unknown population mean and let 2 be the unknown population variance. Let X1,…,X100be the 100 amperages of the sample batteries. The observed value of the sample mean is 185.5. Since is the mean of a large sample, and the Central Limit Theorem specifies that it comes from a normal distribution with mean  and whose standard deviation is .

  12. Computing a 95% Confidence Interval The 95% confidence interval (CI)is . So, a 95% CI for the mean is 185.5  1.96 (5/√100) or 185.5  0.98 or (184.52, 186.48). We can use the sample standard deviation as an estimate for the population standard deviation, since the sample size is large. We can say that we are 95% confident, or confident at the 95% level, that the population mean amperage lies between 184.52 and 186.48. Warning: The methods described here require that the data be a random sample from a population. When used for other samples, the results may not be meaningful.

  13. Illustration of Capturing True Mean Here is a normal curve, which represents the distribution of . The middle 95% of the curve, extending a distance of 1.96 on either side of the population mean , is indicated. The following illustrates what happens if lies within the middle 95% of the distribution:

  14. Illustration of Not Capturing True Mean If the sample mean lies outside the middle 95% of the curve: Only 5% of all the samples that could have been drawn fall into this category. For those more unusual samples the 95% confidence interval fails to cover the true population mean .

  15. Question? • Does this 95% confidence interval actually cover the population mean ? • It depends on whether this particular sample happened to be one whose mean came from the middle 95% of the distribution or whether it was a sample whose mean was unusually large or small, in the outer 5% of the population. • There is no way to know for sure into which category this particular sample falls. • In the long run, if we repeated these confidence intervals over and over, then 95% of the samples will have means in the middle 95% of the population. Then 95% of the confidence intervals will cover the population mean.

  16. Pieces of CI • Recall that the CI was 185.5  0.98. • 185.5 was the sample mean which is a point estimate for the population mean. • We call the plus-or-minus number 0.98 the margin of error • The margin of error is the product of 1.96 and = 0.5. • We refer to which is the standard deviation of , as the standard error. • In general, the standard error is the standard deviation of the point estimator. • The number 1.96 is called the critical valuefor the confidence interval. The reason that 1.96 is the critical value for a 95% CI is that 95% of the area under the normal curve is within – 1.96 and 1.96 standard errors of the population mean.

  17. Other CI Levels • Suppose we are interested in 68% confidence intervals, then we know that the middle 68% of the normal distribution is in an interval that extends 1.0 on either side of the population mean. • It follows that an interval of the same length around specifically, will cover the population mean for 68% of the samples that could possibly be drawn. • For our example, a 68% CI for the diameter of pistons is 185.5  1.0(0.5), or (185.0, 186.0).

  18. 100(1 - )% CI Let X1,…,Xn be a large (n > 30) random sample from a population with mean  and standard deviation , so that is approximately normal. Then a level 100(1 - )% confidence interval for  is where . When the value of  is unknown, it can be replaced with the sample standard deviation s.

  19. 100(1 - )% CI

  20. Specific Intervals for  • is a 68% interval for . • is a 90% interval for . • is a 95% interval for . • is a 99% interval for . • is a 99.7% interval for .

  21. More About CI’s • The confidence level of an interval measures the reliability of the method used to compute the interval. • A level 100(1 - )% confidence interval is one computed by a method that in the long run will succeed in covering the population mean a proportion 1 -  of all the times that it is used. • In practice, there is a decision about what level of confidence to use. • This decision involves a trade-off, because intervals with greater confidence are less precise.

  22. Probability vs. Confidence • In computing CI, such as the one of amperage of batteries: (184.52, 186.48), it is tempting to say that the probability that  lies in this interval is 95%. • The term probability refers to random events, which can come out differently when experiments are repeated. • 184.52 and 186.48 are fixed, not random. The population mean is also fixed. The mean diameter is either in the interval or not. • There is no randomness involved. • So, we say that we have 95% confidence that the population mean is in this interval.

  23. Probability vs. Confidence • 68% CI • 95% CI • 99.7% CI

  24. Determining Sample Size • In Example 5.4, a 95% CI was given by 12.68  1.89. • This interval specifies the mean to within  1.89. Now assume that the interval is too wide to be useful. • Assume that it is desirable to produce a 95% confidence interval that specifies the mean to within  0.5. • To do this, the sample size must be increased. • The width of a CI is specified by • If the desired with is  w then • Solving this equation for n yields

  25. One-Sided Confidence Intervals • We are not always interested in CI’s with both an upper and lower bound. • For example, we may want a confidence interval on battery life. We are only interested in a lower bound on the battery life. • With the same conditions as with the two-sided CI, the level 100(1-)% lower confidence bound for  is and the level 100(1-)% upper confidence bound for  is

  26. Small Sample CIs for a Population Mean • The methods that we have discussed for a population mean previously require that the sample size be large. • When the sample size is small, there are no general methods for finding CI’s. • If the population is approximately normal, a probability distribution called the Student’s t distribution can be used to compute confidence intervals for a population mean.

  27. More on CI’s • What can we do if is the mean of a small sample? • If the sample size is small, s may not be close to , and may not be approximately normal. If we know nothing about the population from which the small sample was drawn, there are no easy methods for computing CI’s. • However, if the population is approximately normal, it will be approximately normal even when the sample size is small. It turns out that we can use the quantity , but since s may not be close to , this quantity has a Student’s t distribution.

  28. Student’s t Distribution • Let X1,…,Xnbe a small (n < 30) random sample from a normal population with mean . Then the quantity has a Student’s t distribution with n -1 degrees of freedom (denoted by tn-1). • When n is large, the distribution of the above quantity is very close to normal, so the normal curve can be used, rather than the Student’s t.

  29. More on Student’s t • The probability density of the Student’s t distribution is different for different degrees of freedom. • The t curves are more spread out than the normal. • Table C, called a tDistribution, provides probabilities associated with the Student’s t distribution.

  30. Student’s t CI Let X1,…,Xnbe a small random sample from a normal population with mean . Then a level 100(1 - )% CI for  is To be able to use the Student’s t distribution for calculation and confidence intervals, you must have a sample that comes from a population that is approximately normal. Samples such as these rarely contain outliers. So if a sample contains outliers, this CI should not be used.

  31. Other CI’s Let X1,…,Xnbe a small random sample from a normal population with mean . • Then a level 100(1 - )% upper confidence bound for  is • Then a level 100(1 - )% lower confidence bound for  is • Occasionally a small sample may be taken from a normal population whose standard deviation  is known. In these cases, we do not use the Student’s t curve, because we are not approximating  with s. The CI to use here is the one using the z table, which we discussed in the first section.

  32. 3.8.4 Tolerance Intervals • A confidence interval for a parameter is an interval that is likely to contain the true value of the parameter. • Prediction and tolerance intervals are concerned with the population itself and with values that may be sampled from it in the future. • These intervals are only useful when the shape of the population is known, here we assume the population is known to be normal.

  33. Prediction Interval • A prediction interval is an interval that is likely to contain the value of an item that will be sampled from the population at a future time. • We “predict” that a value that is yet to be sampled from the population will fall within the predication interval.

  34. 100(1 – α)% Prediction Interval Let X1,…,Xnbe a random sample from a normal population. Let Y be another item to be sampled from this population, whose value has not yet been observed. The 100(1 – α)% prediction interval for Y is The probability is 1 – α that the value of Y will be contained in this interval. One sided intervals may also be constructed.

  35. Comparing CI and PI • The formula for the PI is similar to the formula for the CI of a mean of normal population. • The prediction interval has a small adjustment to the standard error with the additional + 1 under the square root. • This reflects the random variation in the value of the sampled item that is to be predicted. • Prediction intervals are sensitive to the assumption that the population is normal. • If the shape of the population differs much from the normal curve, the prediction interval may be misleading. • Large samples do not help, if the population is not normal then the prediction interval is invalid.

  36. Tolerance Intervals • A tolerance interval is an interval that is likely to contain a specified proportion of the population. • First assume that we have a normal population whose mean μ and standard deviation σ are known. • To find an interval that contains 90% of the population, we have μ± 1.645σ. • In general, the interval μ± zγ/2σ will contain 100(1 – γ)% of the population. • In practice, we do not know μ or σ. Instead we use the sample mean and sample standard deviation.

  37. Consequences • Since we are estimating the mean and standard deviation from the sample, • We must make the interval wider than it would be if μ and σ were known. • We cannot be 100% confident that the interval actually contains the required proportion of the population.

  38. Construction of Tolerance Interval • We must specify the proportion 100(1 – γ)% of the population that we wish the interval to contain. • We must also specify the confidence 100(1 – α)% that the interval actually contains the specified proportion. • It is then possible to find a number kn,α,γsuch that the interval will contain at least 100(1 – γ)% of the population with confidence 100(1 – α)%. Values of kn,α,γ are presented in various statistical books (for example Table A.4 in Principles of Statistics for Engineers and Scientists)

  39. Tolerance Interval Summary Let X1,…,Xnbe a random sample from a normal population. A tolerance interval for containing at least 100(1 – γ)% of the population with confidence 100(1 – α)% is Of all the tolerance intervals that are computed by this method, 100(1 – α)% will actually contain at least 100(1 – γ)% of the population.

  40. 3.8.4 Hypothesis Tests • Formulate a hypothesis that is to be tested • Use the data to test the hypothesis • Determine whether or not the hypothesis should be rejected • The hypothesis that is being tested is called the null hypothesis (H0), which is tested against an alternative hypothesis (H1). • Hypothesis tests are used implicitly when control charts are employed.

  41. Hypothesis Tests • Hypothesis tests using t or z are of the form Where:  is the parameter to be estimated is the point estimator is the estimated standard deviation of the point estimator

  42. Hypothesis Tests • In general, hypotheses that are tested are hardly ever true. • A hypothesis that is being tested is more likely to be rejected for a very large sample size than for a very small sample size.

  43. P-Value • The P-value measures the plausibility of H0. • The smaller the P-value, the stronger the evidence is against H0. • If the P-value is sufficiently small, we may be willing to abandon our assumption that H0 is true and believe H1 instead. • This is referred to as rejecting the null hypothesis.

  44. Steps in Performing a Hypothesis Test • Define H0 and H1. • Assume H0 to be true. • Compute a test statistic. A test statistic is a statistic that is used to assess the strength of the evidence against H0. A test that uses the z-score as a test statistic is called a z-test. • Compute the P-value of the test statistic. The P-value is the probability, assuming H0 to be true, that the test statistic would have a value whose disagreement with H0 is as great as or greater than what was actually observed. The P-value is also called the observed significance level.

  45. One and Two-Tailed Tests • When H0 specifies a single value for , both tails contribute to the P-value, and the test is said to be a two-sided or two-tailed test. • When H0 specifies only that  is greater than or equal to, or less than or equal to a value, only one tail contributes to the P-value, and the test is called a one-sidedor one-tailed test.

  46. Drawing Conclusions from the Results of Hypothesis Tests • There are two conclusions that we draw when we are finished with a hypothesis test, • We reject H0. In other words, we concluded that H0 is false. • We do not reject H0. In other words, H0 is plausible. • One can never conclude that H0 is true. We can just conclude that H0 might be true. • We need to know what level of disagreement, measured with the P-value, is great enough to render the null hypothesis implausible.

  47. More on the P-value • The smaller the P-value, the more certain we can be that H0 is false. • The larger the P-value, the more plausible H0 becomes but we can never be certain that H0 is true. • A rule of thumb suggests to reject H0 whenever P 0.05. While this rule is convenient, it has no scientific basis.

  48. Comments • Some people report only that a test significant at a certain level, without giving the P-value. Such as, the result is “statistically significant at the 5% level.” • This is poor practice. • First, it provides no way to tell whether the P-value was just barely less than 0.05, or whether it was a lot less. • Second, reporting that a result was statistically significant at the 5% level implies that there is a big difference between a P-value just under 0.05 and one just above 0.05, when in fact there is little difference. • Third, a report like this does not allow readers to decide for themselves whether the P-value is small enough to reject the null hypothesis. • Reporting the P-value gives more information about the strength of the evidence against the null hypothesis and allows each reader to decide for himself or herself whether to reject the null hypothesis.

  49. Comments on P Let  be any value between 0 and 1. Then, if P  , • The result of the test is said to be significantly significant at the 100% level. • The null hypothesis is rejected at the 100% level. • When reporting the result of the hypothesis test, report the P-value, rather than just comparing it to 5% or 1%.

  50. Significance • When a result has a small P-value, we say that it is “statistically significant.” • In common usage, the word significant means “important.” • It is therefore tempting to think that statistically significant results must always be important. • Sometimes statistically significant results do not have any scientific or practical importance.

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