Estimation of Mean (CI and HT with visuals) and Sample Size 2023(1)

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Nov 24, 2024

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CALCULATOR FOR ONE SAMPLE TESTING for µ (Z or t-test) Sample Parameters Is Sigma known and n>30 No Enter Sample Size 20 Enter Sample Average (Xbar) 72.3 Enter sample standard deviation (s) 9.521 Approximate Standard Error 2.129 Hypothesis Testing and CI Estimation Enter ⍺ 0.05 67.5 Sample t statistic 2.255 Sample t statistic 2.255 Critical t 2.093 p-value 0.036 Decision Reject Ho Enter hypothesized mean μ o -6 -4 -2 0 2 54 56 56 2-sided Testing (Ha: µ≠µo)

How to interpret these graphics: The graphics shown here demonstrate the distribution u Understanding t-critical t-critical represents the boundary of Ho if the true popu The critical t values are represented by the solid red line the rejection of a one sided test is . ⍺ For example, say =0.05, the probability to the lef and r ⍺ If t-sample (- - - dashed black line ) falls outside the boun Understanding p-value The p-value represents a conditional probability and it i demonstrated in the graphics above allowing us to calcu expressed as P(observing our results or more extreme|H If the p-value is ≦ ⍺ , we reject Ho in favor of Ha. For exa The p-value represents the probability of a Type I error o The more extreme our t-sample in the direction of the a Understanding the decision to "reject Ho in favor of Ha" If we reject Ho, we are essentially saying that if the null h alternative to be true. However, we DON'T definitively k

Hypothesis Testing Test Ha Critical t 95% Confidence Interval p-value Decision 2-sided μ≠67.5 2.093 (67.844,76.756) 0.036 Reject Ho Lef-sided μ<67.5 -1.729 (- ∞,75.981) 0.982 Cannot reject Ho Right sided μ>67.5 1.729 (68.619,+ ∞) 0.018 Reject Ho Sample t statistic 2.255 Critical t -1.729 p-value 0.982 Decision Cannot reject Ho 4 6 -6 -4 -2 0 2 4 54 56 Lef sided Testing (Ha: µ<µo)

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under the null hypothesis meaning that each of these distributions is centered around a ulation mean equals µ o . e in the graphic. A 2 sided test has 2 boundaries and a 1 sided test has only 1 boundary. right of t-critical in a 2 sided test is 2.5% while in the lef sided test, the probability to th ndary(ies) of Ho, we can reject Ho in favor of Ha. is cumulative. It is a conditional probability because it is calculated under the assumptio ulate probabilities. The p-value can be expressed as the cumulative probability of obser Ho is true). ample, if ⍺ =0.05 and the p-value is 0.02, we can reject Ho in favor of Ha at the "5% sign or the probability of rejecting the null hypothesis if the null hypothesis is true. In the exa alternative, the lower the p-value. hypothesis is true, the probability of observing our test results (or more extreme) is so l know if Ho or Ha is true. In fact, if the p-value is 0.02 and we reject Ho, there is a 2% ch

Reject Ho when Xbar < 63.044 or > 71.956 Reject Ho when Xbar < 63.819 Reject Ho when Xbar > 71.181 Sample t statistic 2.255 Critical t 1.729 p-value 0.018 Decision Reject Ho Threshold for Xbar to reject Ho 6 -6 -4 -2 0 2 56 54 Right sided Testing (Ha: µ>µo) Try out this example... One sample Right sided t-test Variable: Midterm Exam Grade Sample size: 20 Sample mean: 72.3 Sample SD (s): 9.521 Ho: µ 67.5 Ha: µ>67.5 ≦ Test Results t-statistic: 2.255 df: 19 p-value: 0.018 One sided 95% CI: (68.619,+∞)

a Z or t-value =0 when the mean is µ o . . The probability of each rejection zone of a 2-sided test is /2 whereas the probab ⍺ he lef t-critical is 5%. In a right sided test, the probability to the right of t-critical is on that the null hypothesis is true. If the null hypothesis is true, the distribution is rving your sample results or more extreme IF the null hypothesis is true and can b nificance level" or with "95% confidence". ample above, if we reject the null hypothesis, there is a 2% risk that we could be w low ( ) that we believe the null hypothesis is likely not true. Thus, we believe th ≦ ⍺ hance that we could be wrong and the null hypothesis is actually true (Type I error

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4 6

bility of s 5%. e wrong. he ).

Sample Size Calculation for μ ⍺ 0.05 Half-length of interval (margin of error, accuracy) 0.01 Standard deviation (estimate) 0.01 Z multiple (2-sided) 1.96 Sample size 4

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