Point Estimation and Confidence Intervals in Engineering Statistics, Study notes of Statistics

Download Point Estimation and Confidence Intervals in Engineering Statistics and more Study notes Statistics in PDF only on Docsity! Notes ST372: Engineering Statistics Handout 1 Point Estimation One of the most important things we want to be able to do is estimate quantities. In order to do this we perform experiments and collect data. Lets consider the case where we want to estimate the gravitational acceleration on some unknown planet where we have just arrived. We know from basic laws of motion that y(t) = y0 + 1 2 gt2 So we can conceive of an experiment where we drop a mass from a known height y0 and measure the time it takes to fall to the ground. We assume a known height of one meter and observe the following times t = (0.472, 0.462, 0.444, 0.450, 0.460, 0.478, 0.470, 0.439, 0.451, 0.459)′ We can take the sample average as an estimate of t, t̂ = 0.4982. Can we then use some simple algebra to show that t = √ 2y0 g g = 2y0 t2 . We could the transform each observation and calculate the sample mean or we could substitute t̂. In either case we could get a estimate of the value for gravitational acceleration. Method of Moments When we make estimates of parameters like this we have two approaches. One we can use the method of moments approach. This is based on the definition of moments mn = ∫ xnf(x)dx as the nth moment of a density. We know that the distribution mean is the first moment and the variance is the second moment minus the square of the first moment. Knowing this and that the sample moments are defined as m̃n = ∑ x N 1 Notes ST372: Engineering Statistics Handout 1 We can then knowing the mean and variance use the sample moments to solve for the parameters. Consider the gamma distribution f(x) = βα Γ(α) xα−1eβx where E(x) = α β V ar(x) = α β2 After some algebra we can then find the method of moments estimators α̃ = µ2 σ2 β̃ = µ σ2 . Maximum Likelihood Estimators The second method of finding estimators is using the maximum likelihood principle. The likelihood for a given set of data from the distribution f(x) is L(x|θ) = ∏ f(x). The estimators for θ that maximize the likelihood are the maximum likelihood esti- mators. In order to find these estimators we take the derivative of the log-likelihood with respect to θ and set it equal to 0 and solve for θ. As an example consider observations from a normal distribution L(x|θ) = ( 1 2πσ2 )N/2 exp ( −1 2 ∑ (x− µ)2 sigma2 ) the log-likelihood is then −N 2 log(2πσ2)− 1 2 ∑ (x− µ)2 sigma2 the derivative with respect to µ is∑ x−Nµ σ2 = 0 as a result the MLE for µ is ∑ x/N , the same as the moment estimator. This procedure can be difficult as in the case of the gamma distribution, I encourage you to find out for yourselves. 2 Notes ST372: Engineering Statistics Handout 1 based on the normal density but has heavier tails, as it accounts for the uncertainty in using the estimate s2. The resulting 100(1− α)% Confidence Interval is x̄± tν,α/2σ/ √ n where ν = N − 1. The values for tν,α/2 are available in tables. The final case we consider is finding a 100(1 − α)% Confidence Interval for a population proportion. In order to do this we note that our experimental data must follow the constraints x > 10 and N − x > 10. Under these conditions we can use the following formula p̂± Zα/2 √ p̂q̂ N where q̂ = 1− p̂. It is important to remember that confidence intervals only give use a measure of uncertainty about the estimation procedure, not about the parameter value. This is an important distinction and one we should keep in mind when using confidence intervals to communicate experimental results. What is the confidence interval for the true time for an object to fall 1 m based on the sample data? What is the confidence interval for g? Hypothesis Tests In science we often have ideas that we want to verify through experimentation. Return to our gravity data from above, let’s assume that we think that the gravitational acceleration for our new planet is less than earth’s gravitation acceleration 9.81m/s2. We can calculate that the true time for an object to fall 1 m on earth is 0.452 seconds. Our data has values more than that but also some less than that. So what can we conclude? Our test procedure has to account for the uncertainty of our observations. To begin this procedure we have to clearly state what we want to test. We need a null hypothesis and an alternative hypothesis. The null hypothesis typically represents the status quo, or the opposite of what we believe. In our case the null 5 Notes ST372: Engineering Statistics Handout 1 hypothesis is that µ ≤ 0.452 the opposite of this, or what we believe is the alternative hypothesis is µ > 0.452. We state these as H0 : µ ≤ 0.452 HA : µ > 0.452. In order to perform our test accounting for uncertainty we ask the question “How likely (or unlikely) are the data we observe given that our null hypothesis is true?” We can answer this question if we know the underlying distribution or we can use what we know about the distribution of x̄ to calculate this probability as P (x > x̄|µ = µ0) noting that x̄ ∼ N(µ, σ2/N). Here note that µ0 is the mean from H0. If we know the variance σ2, or N > 30 we can use the normal tables to find P ( x̄− µ0 σ/n > Z ) . The quantity (x̄−µ0)/σ/n is the test statistic z In the case of our data this becomes P (1.77 > Z) = 0.038 We are going to assume that this works even though we don’t know σ and N < 30. In this case we actually see that there is a pretty small chance that we would have observed this data given that the gravitational acceleration on this planet was the same or greater than earth’s. As a result we can reject H0 and accept HA. This probability is called the p-value. Can we use the same procedure to test g directly? In cases where N < 30 we have to use the t distribution much like we did for confidence intervals. Hypothesis tests are one of three patterns: Two-tailed tests H0 : µ = µ0 HA : µ 6= µ0 6 Notes ST372: Engineering Statistics Handout 1 Upper-tailed test H0 : µ ≤ µA HA : µ > µ0 Lower-tailed test H0 : µ ≥ µA HA : µ < µ0 These are called tailed tests because they refer to how to calculate the p-values. In the case of Lower tailed tests, the p-values is the area under the standard normal curve to the left of the test statistic, for upper tailed test it is the area to the right of the test statistic. In the two tailed case it is the area in tails defined by the test statistic z and −z. There are two types or errors that can be made in this test procedure. α = Type I Error = P (Reject H0|H0 is True) β = Type II Error = P (Fail to Reject H0|H0 is False) The quantity 1− β is P (Reject H0|H0 is False) this quantity is called the power of a test and depends on the sample size, the variance and the true value of µ. Typically we reject H0 when the p-value is less than 0.05. • For hypothesis tests about the mean µ when N > 30 or the variance is known we use the test statistic z = x̄− µ0 σ/ √ N where z ∼ N(0, 1) • When N < 30 and the variance is unknown we use the test statistic t t = x̄− µ0 s/ √ N where t ∼ tν , where ν = N − 1. 7