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REVIEW Hypothesis Tests of Means

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5 Steps for Hypothesis Testing Test Value Method 1.Develop null and alternative hypotheses 2.Specify the level of significance, 3.Use the level of significance to determine the critical values for the test statistic and state the rejection rule for H 0 4.Collect sample data and compute the value of test statistic 5.Use the value of test statistic and the rejection rule to determine whether to reject H 0

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When to use z and When to use t When to use z and When to use t z and t distributions are used in hypothesis testing. _ These are determined by the distribution of X. USE z Large n or sampling from a normal distribution Large n or sampling from a normal distribution σ is known σ is known USE t Large n or sampling from a normal distribution Large n or sampling from a normal distribution σ is unknown σ is unknown

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General Form of Test Statistics for Hypothesis Tests vA test statistic is nothing more than a measurement of how far away the observed value from your sample is from some hypothesized value, v. –It is measured in terms of standard errors z-statistic –σ known = z-statistic with standard error = t-statistic –σ unknown = t-statistic with standard error = The general form of a test statistic is: Depending on whether or not σ is known

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Example The average cost of all required texts for introductory college English courses seems to have gone up substantially as the professors are assigning several texts. –A sample of 41 courses was taken –The average cost of texts for these 41 courses is $86.15 Can we conclude the average cost: 1.Exceeds $80? 2.Is less than $90? 3.Differs from last year’s average of $95? 4.Differs from two year’s ago average of $78?

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Assume the standard deviation is $22. Because the sample size > 30, it is not necessary to assume that the costs follow a normal distribution to determine the z-statistic. In this case because it is assumed that σ is known (to be $22), these will be z-tests. CASE 1: z-tests for σ Known CASE 1: z-tests for σ Known

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Example 1: Can we conclude µ > 80? H 0 : µ = 80 H A : µ > 80 Select α =.05 TEST: Reject H 0 (Accept H A ) if z > z.05 = 1.645 z calculation: Conclusion: 1.790 > 1.645 There is enough evidence to conclude µ > 80. 1 2 3 4 5

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Example 2: Can we conclude µ < 90? H 0 : µ = 90 H A : µ < 90 Select α =.05 TEST: Reject H 0 (Accept H A ) if z <-z.05 = -1.645 z calculation: Conclusion: -1.121 > -1.645 There is not enough evidence to conclude µ < 90. 1 2 3 4 5

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Example 3: Can we conclude µ ≠ 95? H 0 : µ = 95 H A : µ ≠ 95 Select α =.05 TEST: Reject H 0 (Accept H A ) if z z.025 = 1.96 z calculation: Conclusion: -2.578 < -1.96 There is enough evidence to conclude µ ≠ 95. 1 2 3 4 5

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Example 4: Can we conclude µ ≠ 78? H 0 : µ = 78 H A : µ ≠ 78 Select α =.05 TEST: Reject H 0 (Accept H A ) if z z.025 = 1.96 z calculation: Conclusion: 2.372 > 1.96 There is enough evidence to conclude µ ≠ 78. 1 2 3 4 5

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P-values are a very important concept in hypothesis testing. p-valueA p-value is a measure of how sure you are that the alternate hypothesis H A, is true. –The lower the p-value, the more sure you are that the alternate hypothesis, the thing you are trying to show, is true. So p-valueα –A p-value is compared to α. If the p-value < α; accept H A – you proved your conjecture If the p-value > α; do not accept H A – you failed to prove your conjectureP-values Low p-values Are Good!

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5 Steps for Hypothesis Testing P-value Method 1.Develop null and alternative hypotheses 2.Specify the level of significance, 3.Collect sample data and compute the value of test statistic 4.Calculate p-value: Determine the probability for the test statistic 5.Compare p-value and : Reject H 0 (Accept H A ), if p-value <

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Calculating p-values p-valueA p-value is the probability that, if H 0 were really true, you would have gotten a value as least as great as the sample value for “>” tests at most as great as the sample value for “<” tests at least as far away from the sample value for “≠” tests z-valueFirst calculate the z-value for the test. p-valueThe p-value is calculated as follows: TESTP-valueEXCEL “>”P(Z>z) – Area to the right of z=1-NORMSDIST(z) “<”P(Z<z) – Area to the left of z=NORMSDIST(z) “≠” For z < 0: For z < 0: 2*(Area to the left of z) For z > 0: For z > 0: 2*(Area to the right of z) =2*NORMSDIST(z) =2*(1-NORMSDIST(z))

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0 Z v P-Value for “≠” Test, With z>0 0 Z v P-Value for “≠” Test, With z<0 z 0 Z v P-Value for “>” Test z P-value v P-Value for “<” Test 0 Z z P-value P-value = 2*area z

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Examples – p-Values Examples – p-Values >Example 1: Can we conclude µ > 80? z = 1.79 P-value.0367P-value = 1 -.9633 =.0367 (< α =.05). Can Can conclude µ > 80. <Example 2: Can we conclude µ < 90? z = -1.12 P-value.1314P-value =.1314 (> α =.05). Cannot Cannot conclude µ < 90. ≠Example 3: Can we conclude µ ≠ 95? z = -2.58 P-value.0098P-value = 2(.0049) =.0098 (< α =.05). Can Can conclude µ ≠ 95. ≠Example 4: Can we conclude µ ≠ 78? z = 2.37 P-value.0178P-value = 2(1-.9911) =.0178 (< α =.05). Can Can conclude µ ≠ 78.

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=AVERAGE(A2:A42)

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=(D4-D7)/(D1/SQRT(D2)) =1-NORMSDIST(D8)

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=(D4-D12)/(D1/SQRT(D2)) =NORMSDIST(D13)

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=(D4-D17)/(D1/SQRT(D2)) =2*NORMSDIST(D18)

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=(D4-D22)/(D1/SQRT(D2))=2*(1-NORMSDIST(D23))

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Because the sample size > 30, it is not necessary to assume that the costs follow a normal distribution to determine the t-statistic. In this case because it is assumed that σ is unknown, these will be t-tests with 41-1 = 40 degrees of freedom. Assume s = 24.77. CASE 2: t-tests for σ Unknown CASE 2: t-tests for σ Unknown

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Example 1: Can we conclude µ > 80? H 0 : µ = 80 H A : µ > 80 Select α =.05 TEST: Reject H 0 (Accept H A ) if t >t.05,40 = 1.684 t calculation: Conclusion: 1.590 < 1.684 Cannot conclude µ > 80. 1 2 3 4 5

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Example 2: Can we conclude µ < 90? H 0 : µ = 90 H A : µ < 90 Select α =.05 TEST: Reject H 0 (Accept H A ) if t<-t.05,40 = -1.684 t calculation: Conclusion: -0.995 > -1.684 Cannot conclude µ < 90. 1 2 3 4 5

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Example 3: Can we conclude µ ≠ 95? H 0 : µ = 95 H A : µ ≠ 95 Select α =.05 TEST: Reject H 0 (Accept H A ) if t t.025,40 = 2.021 t calculation: Conclusion: -2.288 < -2.021 Can conclude µ ≠ 95. 1 2 3 4 5

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Example 4: Can we conclude µ ≠ 78? H 0 : µ = 78 H A : µ ≠ 78 Select α =.05 TEST: Reject H 0 (Accept H A ) if t t.025,40 = 2.021 t calculation: Conclusion: 2.107 > 2.012 Can conclude µ ≠ 78. 1 2 3 4 5

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The TDIST Function in Excel The TDIST Function in Excel TDIST(t,degrees of freedom,1)TDIST(t,degrees of freedom,1) gives the area to the right of a positive value of t. –1-TDIST(t,degrees of freedom,1) gives the area to the left of a positive value of t. –Excel does not work for negative vales of t. –But the t-distribution is symmetric. Thus, The area to the left of a negative value of t = area to the right of the corresponding positive value of t. TDIST(-t,degrees of freedom,1) gives the area to the left of a negative value of t. 1-TDIST(-t,degrees of freedom,1) gives the area to the right of a negative value of t. TDIST(t,degrees of freedom,2)TDIST(t,degrees of freedom,2) gives twice the area to the right of a positive value of t. –TDIST(-t,degrees of freedom,2) –TDIST(-t,degrees of freedom,2) gives twice the area to the right of a negative value of t.

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p-Values for t-Tests Using Excel P-values for t-tests are calculated as follows: H A TEST Sign of t EXCELP-value “>”>0<0 Usual case =TDIST(t,degrees of freedom,1) Usual case =1-TDIST(-t,degrees of freedom,1) “<”<0>0 Usual case =TDIST(-t,degrees of freedom,1) Usual case =1-TDIST(t,degrees of freedom,1) “≠”<0>0 =TDIST(-t,degrees of freedom,2) =TDIST(t,degrees of freedom,2)

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=(D3-G2)/D4 =TDIST(G3,40,1)

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=(D3-G7)/D4 =TDIST(-G8,40,1)

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=(D3-G12)/D4 =TDIST(-G13,40,2)

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=(D3-G17)/D4 =TDIST(G18,40,2)

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Test Value vs P-Value Example of one-tailed, positive test value is known is unknown Test Value Compare z test with z critical Accept H A if z test > z critical Compare t test with t critical Accept H A if t test > t critical P- value Compare p-value (based on normal distribution) with α Accept H A if p-value < α Compare p-value (based on t distribution) with α Accept H A if p-value < α

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Review When to use z and when to use t in hypothesis testing –σ known – z –σ unknown – t z and t statistics measure how many standard errors the observed value is from the hypothesized value Form of the z or t statistic Meaning of a p-value z-tests and t-tests –By hand –Excel

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