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Advanced Math Topics

Advanced Math Topics. Finals Review: Chapters 12 & 13. A teacher surveyed his students to see if they would take a class from him again. He organized the results by G.P.A. in the table below:. Is there a significant difference between the answers for each G.P.A.? Use 5% level of significance.

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Advanced Math Topics

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  1. Advanced Math Topics Finals Review: Chapters 12 & 13

  2. A teacher surveyed his students to see if they would take a class from him again. He organized the results by G.P.A. in the table below: Is there a significant difference between the answers for each G.P.A.? Use 5% level of significance. We can use the Chi-square statistic to answer the question.

  3. E = 0.5(50) = (25) E = 0.5(80) = (40) E = 0.5(20) = (10) The two E values in each column must add to the total E = 25 E = 40 E = 10 Total: 50 80 20 sum of the top row 75 The null hypothesis is that the proportions of yes/no responses is not different between the columns. p = = = 0.5 total sample size 150 (O – E)2 (28 – 25)2 = 0.36 Χ2 = Σ = E 25 Do this for all six boxes and sum them up. Χ2 = 0.36 + 0.40+ 0.10 + 0.36 + 0.40 + 0.10 = 1.72 Find the critical value using d.f. = 3-1 = 2 and look under the 0.05 column Χ2 = 5.991 Since our test statistic (1.72) is less than the critical value (5.991), we accept the null hypothesis. There is not a significant difference between the responses of students with different GPA’s.

  4. 1) The number of phone calls received per day by a local company is as follows: E = 169.4 E = 169.4 E = 169.4 E = 169.4 E = 169.4 Using a 5% level of significance, test the null hypothesis that the number of calls received is independent of the day of the week. We first calculate the number of expected calls per day. If the number of calls per day is independent of the day of the week, we would expect to receive… 173 + 153 + 146 + 182 + 193 (O – E)2 Χ2 = Σ = 169.4 E 5 (173 – 169.4)2 (153 – 169.4)2 (146 – 169.4)2 (182 – 169.4)2 (193 – 169.4)2 = + + + + = 169.4 169.4 169.4 169.4 169.4 Find the critical value using d.f. = 5-1 = 4 and look under the 0.05 column 9.1216 Χ2 = 9.488 Since our test statistic (9.1215) is less than the critical value (9.488), we accept the null hypothesis. The phone calls received are independent of the day of the week.

  5. 2) A scientist claims that when two rats mate, the offspring will be black, gray, and white in the proportion 5:4:3. Out of 180 newborn rats, 71 are black, 69 are gray, and 40 are white. Can we accept the scientist’s claim? Use a 5% level of significance. The expected probability of a black rat is 5/12. The expected probability of a gray rat is 4/12. 5 + 4 + 3 = 12 The expected probability of a white rat is 3/12. The expected frequency of a black rat is 180(5/12) = 75. The expected frequency of a gray rat is 180(4/12) = 60. The expected frequency of a white rat is 180(3/12) = 45. (O – E)2 Χ2 = Σ E (71 – 75)2 (69 – 60)2 (40 – 45)2 5.991 2.1189 vs. = + + = 75 60 45 We accept the scientist’s claim!

  6. 1) Is eye color independent of hair color? Use a 5% level of significance. Total 43 E = 43(54)/100 E = 43(46)/100 E = 23.22 E = 19.78 57 E = 57(54)/100 E = 57(46)/100 E = 30.78 E = 26.22 Total 54 46 100 (O – E)2 (10 – 23.22)2 = 7.527 Χ2 = Σ = E 23.22 Since our test statistic is larger, eye color is dependent on hair color. Do this for all four boxes and sum them up. 3.841 Χ2 = 7.527 + 8.836+ 5.678 + 6.665 = 28.706 vs. The degrees of freedom is (# of rows – 1)(# of columns – 1). (2 – 1)(2 – 1) = 1

  7. 4) A study wants to compare the cost of hospitals across the country. The prices show the daily hospital charge from four hospitals in each region. At a 5% level, test the claim that the average daily charge is significantly different depending on the region. To solve this type of question, use an ANOVA table.

  8. An ANOVA Table Degrees of freedom Sum of Squares Mean Square f-ratio Σ(row total2) ___Total2__ # of Boxes - Factor of the Experiment r – 1 r = # of rows # of columns Cell 1/Cell 4 Cell 7/Cell 8 r(c – 1) c = # of columns Σ(row total2) (sum of each #2) - Cell 2/Cell 5 Error # of columns Add cells 1 & 2 Add cells 4 & 5 Total Compare your test statistic to the critical value. This is found on A14-15(5%) or A16-17(1%) by looking up the d.f. for the numerator and denominator from Cell 9.

  9. Prepare your green notecard(put all formulas on the card, no formulas will be given) and study Chapters 11, 12 and 13 for the Final tomorrow. A good way to study is to look at the Powerpoints from each Chapter, or the Review Powerpoints, and solve each question…then reveal the answer. Bring in your textbooks tomorrow and sign up for Schoolloop for homework.

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