Survey

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Chapter 9 Tests of Significance Target Goal: I can perform a significance test to support the alternative hypothesis. I can interpret P values in context. 9.1a Testing a Claim h.w: pg. 546: 1-7 odd Tests of Significance: to test a claim about an unknown population parameter. Example : Sweetening Colas • Diet colas use artificial sweeteners to avoid sugar. Colas with artificial sweeteners gradually lose sweetness over time. Manufacturers therefore test new colas for loss of sweetness before marketing them. Trained tasters sip the cola along with drinks of standard sweetness and score the cola on a “sweetness score” of 1 to 10. The cola is then stored for a period of time, then each taster scores the stored cola. • This is a matched pairs experiment. • The reported data is the difference in tasters’ scores. The bigger the difference, the bigger the loss in sweetness. 2.0 0.4 0.7 2.0 -0.4 2.2 -1.3 1.2 1.1 2.3 Positive score: lost sweetness Negative score: gained sweetness • The sample mean indicates a small loss of sweetness. x 1.02 • Consider that a different sample of tasters would have resulted in different scores, and that some variation in scores is expected due to chance. Does the data provide good evidence that the cola lost sweetness in storage? Perform a Significance Test. Step 1: State - Identify the population parameter. State the null and alternative hypothesis. • The parameter of interest is μ, the mean loss in sweetness of all colas produced by the manufacturer. . There is no effect or change in the population. This is the statement we are trying to find evidence against. • The cola does not lose sweetness. H 0 : 0 “ H- nought or null hypothesis” There is an effect or change in the population. • This is the statement we are trying to find evidence for. • The cola does lose sweetness. H a : 0 alternative hypothesis Step 2: Plan Choose the appropriate inference procedure. Verify the conditions for using the selected procedure. • We will be more specific about this later. Step 3: Do - Calculate a statistic to estimate the parameter. Is the value of the statistic far from the value of the parameter? • If so, reject the null hypothesis. • If not, accept the null hypothesis. Calculate the test statistic. • Suppose the individual tasters’ scores vary according to a normal distribution with mean μ and σ = 1. • We want to test the null hypothesis so we assume μ = 0. • The sampling model for x is approximately normal with mean 0 and standard deviation 1 n 10 0.316 N(0, 0.316) Compare one cola with x = 0.3 and our cola with x = 1.02 to show what it means in terms of H 0 : 0 . -0.9 -0.6 -0.3 0 0.3 0.6 0.9 The cola with 0.3 could happen just by chance. Our cola is so far out on the normal curve that it is good evidence that this cola did lose sweetness. Reject the null hypothesis. Our sample mean, x , was 1.02. • Assuming that the null hypothesis is true, what is the probability of getting a result at least that large? Calculate the P-value. (the probability of the observed x ) • If P-value is small, your result is statistically significant. Determine P-value • Normalcdf ( 1.02, 1E99, 0, 0.316 ) = 0.0006 • The probability to the right of x is called the P-value. • The P-value is small, so our result is statistically significant. Reject the null hypothesis. 0.0006 1.02 Alternate way to calculate one sided P-Value Find Z-value for N(0, 0.316 ), for sample mean 0.3. x 0 0.3 0 P( x 0.3) P 0.316 0.316 • = P(Z ≥ 0.95) • = 1 - P(Z ≤ 0.95) • = 1 – 0.8289 = 0.1711 Is this good evidence against the null hypothesis? • There is about a 17% chance that we will obtain a sample of 10 sweetness loss values whose mean is 0.3 or greater. • This sample could occur quite easily by chance alone. • This evidence against Ho is not strong. (High P-value) Step 4: Interpret the results in the context of the problem for our original data. x 1.02 • The P-value is small,0.0006 (good evidence against null hypothesis). • meaning that we would only expect to get this result in 6 out of 10,000 samples. • This is very unlikely, so we will reject the null hypothesis in favor of the alternative hypothesis and conclude that the cola actually did lose sweetness. Statistically Significant • If the P-value is small we say that our result is statistically significant. • The smaller the P-value, the stronger the evidence provided by the data. How small is small enough? • Compare the P-value to the value of the significance level α (alpha). This value is usually predetermined. • If the P-value is as small as or smaller than α, we say that the data are statistically significant at level α. Read pg. 527 - 532