2500M_Seury_L24_6pg qr

# 2500M_Seury_L24_6pg qr - Table D[7 Inference for...

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1 [7] Inference for Distributions Lecture Notes Simin Seury , Department of Economics, York University, Canada Date: April 1 2010 The one-sample t -test As in the previous chapter, a test of hypotheses requires a few steps: 1. Stating the null and alternative hypotheses ( H 0 versus H a ) 2. Deciding on a one-sided or two-sided test 3. Choosing a significance level α 4. Calculating t and its degrees of freedom 5. Finding the area under the curve with Table D 6. Stating the P-value and interpreting the result n s x t 0 μ - = One-sided (one-tailed) Two-sided (two-tailed) The P-value is the probability, if H 0 is true, of randomly drawing a sample like the one obtained or more extreme, in the direction of H a . The P-value is calculated as the corresponding area under the curve, one-tailed or two-tailed depending on H a : Table D For df = 9 we only look into the corresponding row. For a one-sided H a , this is the P-value (between 0.01 and 0.02); for a two-sided H a , the P-value is doubled (between 0.02 and 0.04). 2.398 < t = 2.7 < 2.821 thus 0.02 > upper tail p > 0.01 The calculated value of t is 2.7. We find the 2 closest t values. Excel TDIST ( x , degrees_freedom , tails ) TDIST = P( X > x ) for a random variable X following the t distribution ( x positive). Use it in place of Table C or to obtain the p-value for a positive t-value. box4 X is the standardized value at which to evaluate the distribution (i.e., “t”). box4 Degrees_freedom is an integer indicating the number of degrees of freedom. box4 Tails specifies the number of distribution tails to return. If tails = 1, TDIST returns the one-tailed p-value. If tails = 2, TDIST returns the two-tailed p-value. TINV(probability,degrees_freedom) Gives the t-value (e.g., t *) for a given probability and degrees of freedom. box4 Probability is the probability associated with the two-tailed t distribution. box4 Degrees_freedom is the number of degrees of freedom of the t distribution. Sweetening colas Cola manufacturers want to test how much the sweetness of a new cola drink is affected by storage. The sweetness loss due to storage was evaluated by 10 professional tasters (by comparing the sweetness before and after storage): Taster Sweetness loss box4 1 2.0 box4 2 0.4 box4 3 0.7 box4 4 2.0 box4 5 -0.4 box4 6 2.2 box4 7 -1.3 box4 8 1.2 box4 9 1.1 box4 10 2.3 Obviously, we want to test if storage results in a loss of sweetness, thus: H 0 : μ = 0 versus H a : μ > 0 Notice, here we do not know the population parameter σ . square4 The population of all cola drinkers is too large. square4 Since this is a new cola recipe, we have no population data. This situation is very common with real data.

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2 Sweetening colas ( continued ) Is there evidence that storage results in sweetness loss for the new cola recipe at the 0.05 level of significance ( α = 5%)? H 0 : μ = 0 versus H a : μ > 0 (one-sided test) box4 The critical value t α = 1.833.
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