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Inferences on a Population Mean Chapter 8 Inferences on A Population Mean 1 8.0 Introduction Nationally, the average score on the mathematics portion of the SAT was 518 (out of 800) in 2006. Suppose researchers in Utah are interested in evaluating how their state is doing preparing students for the SAT. What questions might the researchers want to answer about the performance of Utah students? What is the population the researchers are interested in? What are the parameters and statistics of interest? Inferences on A Population Mean 2 8.0 Introduction Recall: Statistical inference is the science of deducing properties of an underlying probability distribution from a sample. Inferences on A Population Mean 3 8.0 Introduction There are two main aspects of inference: 1. Estimation: estimation allows us to address questions such as “what was the average SAT score of Utah high school seniors in 2006?” 2. Hypothesis testing: hypothesis testing enables us to assess whether the average score in Utah was different from the average score nationwide. Inferences on A Population Mean 4 8.1 Confidence Intervals Suppose we gather a random sample of 50 Utah high school students who took the SAT in 2006 and found that for this sample, the average math score was 525 and the standard deviation was 25 points. Find c such that P( n (X - μ ) / S ≤ c) = 0.95 |. We can use the value of c to find an interval around X . This interval is a 95% confidence interval for the mean SAT math score for Utah high school students who took the SAT in 2006. A confidence interval for an unknown parameter θ contains a set of plausible values of the parameter. It is associated with a confidence level 1-α, which specifies the probability that the confidence interval actually contains the unknown parameter value. Inferences on A Population Mean 5 Inferences on A Population Mean 6 8.1 Confidence Intervals If the size of the sample, n, is 30 or more or if the underlying data are approximately normally distributed (n can be anything), then the sample mean has a t-distribution and we can use this to conduct inference. A 1-α level confidence interval or two-sided t-interval for population mean μ based on a sample of n observations is tα / 2,n −1s tα / 2,n −1s ⎞ ⎛ ⎟⎟ μ ∈ ⎜⎜ x − ,x + n n ⎠ ⎝ Where x is the sample mean and s is the sample standard deviation. Inferences on A Population Mean 7 8.1 Confidence Intervals tα / 2,n −1s tα / 2,n −1s ⎞ ⎛ ⎟⎟ μ ∈ ⎜⎜ x − ,x + n n ⎠ ⎝ • μ is the true, unknown, value of the population parameter. x • is the observed value of the test statistic and the center of the confidence interval. • The margin of error is given by tα / 2,n −1s n • tα/2 is used since for a (1-α)*100% interval, there must be α/2% not covered by the interval on either side. Inferences on A Population Mean 8 8.1 Confidence Intervals All else being equal, as the confidence level increases, does the length of the interval increase or decrease? All else being equal, as the sample size increases, does the length of the interval increase or decrease? Inferences on A Population Mean 9 8.1 Confidence Intervals One-sided 1-α level confidence intervals or t-intervals for population mean μ based on n observations are given by tα ,n −1s ⎞ ⎛ ⎟⎟ μ ∈ ⎜⎜ − ∞, x + n ⎠ ⎝ and tα ,n −1s ⎞ ⎛ μ ∈ ⎜⎜ x − , ∞ ⎟⎟ n ⎝ ⎠ These provide upper and lower bounds for μ respectively. Again, x is the sample mean and s is the sample standard deviation. Inferences on A Population Mean 10 8.1 Confidence Intervals When the population variance is known, then a 1-α level confidence interval for population mean μ is given by the z-interval zα / 2σ zα / 2σ ⎞ ⎛ ,x + μ ∈⎜ x − ⎟ n n ⎠ ⎝ One-sided z intervals for the population mean μ are zα σ ⎞ ⎛ μ ∈ ⎜ − ∞, x + ⎟ n ⎠ ⎝ and zα σ ⎛ ⎞ ,∞⎟ μ ∈⎜ x − n ⎝ ⎠ Where x denotes the sample mean and σ is the population standard deviation. Inferences on A Population Mean 11 8.1 Confidence Intervals In practice, one-sided confidence intervals are seldom used. Furthermore, the population standard deviation is generally unknown. Thus confidence intervals are usually two-sided tintervals. Inferences on A Population Mean 12 8.1 Confidence Intervals • (1 – α)100% is the confidence level. We are (1 – α)100% confident that the computed interval covers the population mean μ. • α is generally quite small, usually 0.05 or 0.01, so that we have a high level of confidence. Thus 95% and 99% CIs are most common. • The value α is the probability that the confidence interval does not cover μ. We accept some probability that the interval doesn’t cover μ in order to infer something meaningful about the population. Inferences on A Population Mean 13 8.1 Confidence Intervals • A confidence interval for μ represents a set of plausible values for μ that are consistent with the observed data. • The interval is random – μ is fixed!! • Increasing the sample size improves the estimate, thus, all else being equal, as the sample size increases, the confidence interval shortens. Inferences on A Population Mean 14 8.1 Confidence Intervals A sample of 61 bottles of chemical solution is obtained and the solution densities are measured (in lbs. per gallon – ppg). The sample mean and standard deviation are 0.768 ppg and 0.0231 ppg respectively. • Construct a 95% confidence interval for the average solution density. • How many additional bottles should be sampled to construct a 95% confidence interval no longer than 0.01 ppg? • Is it plausible that the average solution density is less than 0.77 ppg? 0.75 ppg? Inferences on A Population Mean 15 8.1 Confidence Intervals A random sample of 16 one-kg sugar packets is obtained and the actual weights of the packets are measured. The sample mean is 1.053 kg and the sample standard deviation is 0.058 kg. • Construct a 99% confidence interval for the average sugar package weight. • Is it plausible that the average weight is 1.025 kg? Inferences on A Population Mean 16 8.2 Hypothesis Testing The weight of the sugar packets is nominally 1 kg. Are the data we observed in our sample consistent with this average? Hypothesis Testing allows us to assess the plausibility of a specific statement or hypothesis. Essentially, we want to address the question “Could the difference between the observed sample mean (1.053 kg) and the “known” population mean (1 kg) be due to chance?” Inferences on A Population Mean 17 8.2 Hypothesis Testing We assess the plausibility of a particular hypothesis about the mean by: 1. Stating hypotheses. 2. Collecting data from a sample and record the value of the sample mean and its standard error. 3. Constructing a test statistic to quantify the difference between what we actually observe and what we expected to observe. 4. Computing the probability of observing the data we did if the initial (null) hypothesis is true (p-value). Inferences on A Population Mean 18 8.2 Hypothesis Testing Null Hypothesis: A statements that the observed difference in what we expected to observe in our experiment and what we actually observed is due to chance. A null hypothesis for a population mean μ designates possible values for μ. e.g. H0: μ=μ0 Alternative Hypothesis: A statement that the observed difference is “real” or not due to chance. The “opposite” of the null hypothesis. An alternative hypothesis for a population mean μ is the “opposite” of the null hypothesis. e.g. HA: μ≠μ0 Inferences on A Population Mean 19 8.2 Hypothesis Testing Two-sided hypotheses: H0: μ=μ0 versus HA: μ≠μ0 One-sided hypotheses: H0: μ≤μ0 versus HA: μ>μ0 or H0: μ≥μ0 versus HA: μ<μ0 The alternative hypothesis is the statement of the suspicions we have about the data. It is sometimes easier to set up the alternative hypothesis first. Inferences on A Population Mean 20 8.2 Hypothesis Testing The weight of the sugar packets is nominally 1 kg. Are the data we observed consistent with this average? H0: μ=1 kg versus HA: μ≠1 kg Inferences on A Population Mean 21 8.2 Hypothesis Testing A test statistic is used to quantify the difference between what we expected to observe and what we actually observed in our data. We know from the previous chapter that the sample mean has a tdistribution when the sample size is sufficiently large or the underlying data are approximately normally distributed. Furthermore, under the null hypothesis, the expected value of this distribution is μ0 . Consider drawing many samples from the data. For each sample we can find a test statistic to quantify the distance of the sample mean x from μ0. If the null hypothesis is true, most sample averages will be “close to” μ0. A value very different from μ0 (hence a test statistic with large magnitude) is unlikely. Therefore when we observe an extreme value, this gives us evidence against the null hypothesis. Inferences on A Population Mean 22 8.2 Hypothesis Testing To test the hypothesis H0: μ=μ0 versus HA: μ≠μ0 based on a sample of n observations with sample mean and x sample standard deviation s, the t-statistic is n ( x − μ0 ) t= s • Find the value of the test statistic for the sugar example. n ( x − μ0 ) 16 (1.053 − 1) t= = = 3.66 s 0.058 Inferences on A Population Mean 23 8.2 Hypothesis Testing The p-value is the probability of obtaining the data we observed or data more extreme (less consistent with H0) if H0 is true. Example: Suppose I toss a coin 10 times and get 9 heads. If we assume the coin is fair, what is the probability that I get 9 or 10 heads (the data observed or something more extreme)? Is that probability small enough to convince you that the coin is not fair? What if I toss the coins 100 times and get 99 heads? A small p-value tells us that our data are unlikely to occur if the null hypothesis is true, thus is provides evidence against the null hypothesis. Inferences on A Population Mean 24 8.2 Hypothesis Testing The p-value for the hypothesis test of H0: μ=μ0 versus HA: μ≠μ0 based on a sample of n observations with sample mean x and sample standard deviation s, is given by p = 2P(X≥|t|) where X has a t-distribution with n-1 degrees of freedom. The p-value for the hypothesis test of H0: μ ≥ μ0 versus HA: μ < μ0 is given by p=P(X ≤ t) where X has a t-distribution with n-1 degrees of freedom. The p-value for the hypothesis test of H0: μ ≤ μ0 versus HA: μ>μ0 is given by p = P(X ≥ t) where X has a t-distribution with n-1 degrees of freedom. Inferences on A Population Mean 25 8.2 Hypothesis Testing A small p-value indicates that the null hypothesis is not plausible. “Small” is defined by the significance level α or size of the test. If p<α we say the null hypothesis is implausible and the results are “statistically significant”. We usually use α=0.05, but this is entirely arbitrary. When the null hypothesis is not plausible (p<α), we “reject the null”. When the null hypothesis is plausible (p>α) we “fail to reject” or accept the null. We cannot prove or disprove the null hypothesis, we only show that it is plausible or not. Statistical significance does not imply practical significance. Inferences on A Population Mean 26 8.2 Hypothesis Testing • Find the p-value for the sugar example and draw conclusions for α=0.05. Using a table, we can find a range of possible values for p: 0.001 < P(X ≥ |t|) < 0.005 ⇒ 0.002 < 2P(X ≥ |t|) < 0.01 Using software, we can find the exact p-value p = 2P(X ≥ |t|) = 2P(X ≥ 3.66) =0.0023 Thus we reject the null hypothesis and conclude that the average weight of the sugar packets is not 1 kg. Inferences on A Population Mean 27 8.2 Hypothesis Testing 1. Set up the null and alternative hypotheses. 2. Sample data from the population and compute the sample mean and its standard error. 3. Compute a test statistic assuming that the null hypothesis is true. 4. Calculate the p-value, the probability of observing our data if the null hypothesis is true. 5. Draw conclusions based on the p-value. Small values (usually < 5%) indicate that the null hypothesis is not plausible. Inferences on A Population Mean 28 8.2 Hypothesis Testing Suppose we gather a random sample of 50 Utah high school students who took the SAT in 2006 and found that for this sample, the average math score was 525 and the standard deviation was 25 points. Is this evidence that the average SAT score for a Utah student is greater than the national average (518)? Inferences on A Population Mean 29 8.2 Hypothesis Testing 1. A random sample of 28 plastic items is obtained, and their breaking strengths are measured. The sample mean is 7.22 units and the sample standard deviation is 0.672 units. Is there evidence that the breaking strength is not 7 units? 2. A company is planning a large telephone survey and is interested in assessing how long it will take. In a short pilot study, 40 people are contacted by telephone and are asked the specified set of questions. The times of these 40 telephone surveys have a mean of 9.39 minutes with a standard deviation of 1.041 minutes. Can the company safely conclude that the telephone surveys will last on average no more than 10 minutes? What are the appropriate null and alternative hypotheses? Inferences on A Population Mean 30 8.2 Hypothesis Testing What is the relationship between hypothesis testing and confidence intervals? The values within the (1 – α)100% t-interval represent values of μ0 for which we would fail to reject H0: μ = μ0 at the significance level α. All of the values outside of the confidence interval are values for which we would reject H0: μ = μ0 at the α-level. Inferences on A Population Mean 31 8.2 Hypothesis Testing Suppose a 95% CI for the average weight of newborn babies in the US based on a sample of size 1000 is (7.025, 7.775). • Would you reject H0: μ = 7.5 lbs using significance level α=0.05? • Would you reject H0: μ = 7 lbs at level α=0.05? At significance level α=0.1? Inferences on A Population Mean 32 8.2 Hypothesis Testing What information does the test statistics give us? What does a p-value mean in terms of the null hypothesis? We conduct a study testing regarding the average GPA of USU students. Testing the null hypothesis H0=2.7, we get a p-value p=0.001. Is it safe to say that we have proven that the average GPA is not 2.7? Explain. Inferences on A Population Mean 33