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# Mark

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OUTPUT PROBLEM 1 Bin | Frequency | Cumulative % | Bin | Frequency | Cumulative % | 0 | 0 | 0.00% | 10 | 8 | 26.67% | 10 | 8 | 26.67% | 20 | 5 | 43.33% | 20 | 5 | 43.33% | 50 | 5 | 60.00% | 30 | 4 | 56.67% | 30 | 4 | 73.33% | 40 | 2 | 63.33% | 40 | 2 | 80.00% | 50 | 5 | 80.00% | 60 | 2 | 86.67% | 60 | 2 | 86.67% | 70 | 2 | 93.33% | 70 | 2 | 93.33% | 80 | 2 | 100.00% | 80 | 2 | 100.00% | 0 | 0 | 100.00% | More | 0 | 100.00% | More | 0 | 100.00% |

OUTPUT PROBLEM 2

OUTPUT PROBLEM 3 12 | | VALUE | STATISTIC | FORMULA | 56 | | 31.03333 | MEAN | =AVERAGE(B2:B31) | 15 | | 26.5 | MEDIAN | =MEDIAN(B2:B31) | 14 | | 15 | MODE | =MODE.MULT(B2:B31) | 34 | | | | | 7 | | 8 | P10 | =PERCENTILE.EXC(B2:B31,0.1) | 78 | | 15 | P37 | =PERCENTILE.EXC(B2:B31,0.37) | 23 | | 45 | P75 | =PERCENTILE.EXC(B2:B31,0.75) | 15 | | | | | 45 | | 10.5 | Q1 | =QUARTILE(B2:B31,1) | 8 | | 26.5 | Q2 | =QUARTILE(B2:B31,2) | 65 | | 44.75 | Q3 | =QUARTILE(B2:B31,3) | 44 | | | | | 10 | | 22.053697 | STD. DEV. | =STDEV.P(B2:B31) | 28 | | | | | 15 | | | | | 45 | | 486.36556 | varP | =VAR.P(B2:B31) | 25 | | 503.13678 | varS | =VAR.S(B2:B31) | 9 | | | | | 33 | | 18.904444 | MAD | =AVEDEV(B2:B31) | 8 | | | | | 77 | | 73 | RANGE | =MAX(B2:B31)-MIN(B2:B31) | 44 | | | | | 5 | | 0.6883169 | SKEWNESS | =SKEW(B2:B31) | 30 | | -0.6854225 | KURTOSIS | =KURT(B2:B31) | 10 | | | | | 60 | | | | | 65 | | | | | 10 | | | | | 41 | | | | |

OUTPUT PROBLEM 4 APPLICATIONS OF STANDARD NORMAL DISTRIBUTION | | | | | | | | | | | | | | | | | 12 | 10 | 9 | 10 | 12 | 15 | 15 | 16 | 15 | | 20 | 22 | 23 | 10 | 12 | 10 | 14 | 16 | 17 | | 18 | 20 | 20 | 21 | 10 | 12 | 23 | 10 | 10 |

MEAN | | | =AVERAGE(B4:J6) | | | 14.888889 | STD. DEV. | | | =STDEV(B4:J6) | | | 4.6021177 | Z20 | | | =STANDARDIZE(20,I9,I10) | | 1.11 | Z10 | | | =STANDARDIZE(10,I9,I10) | | -1.06 | %class scored > 20 | | =1-NORMSDIST(I11) | | | 13.34% | %class scored < 20 | | =NORMSDIST(I11) | | | 86.66% | %class scored bet. 10 and 20 | =NORMSDIST(I11)-NORMSDIST(I12) | 72.26% |

OUTPUT PROBLEM 5 COMPARING MEANS OF DIFFERENT GROUPS | | | | | | | | | | | | | | A company which makes battery operated toy cars claims that its product has a mean lifespan of | 5 years with a standard deviation of 2 years. To established the claim, the company tested a random | sample of 40 batteries and found them to have a average life span of 3 years. Test if the claim mean | is significantly different from the sample mean use æ=0.05 | | | | | | | | | | | | Given: | | Req,d: | | | | | | pop mn = 5 yrs. | (µ) | test Ho: µ = x-bar | | | | | pop sd = 2 | (σ) | | | | | | | alpha = 0.05 (2t) | (α) | | | | | | | sample mn = 3 | (x bar) | | | | | | | sample size = 40 | (n) | | | | | | | | | | | | | | | | H | Ho: µ = x-bar | Ha: µ (5 yrs) < > x-bar (3 yrs) | | | | A | use α = 0.05 (2-tailed test) | | | | | | T | since n> 30, 2-groups, compare means - use z-test | | | | | | | | | | | | | C4) Calculate for the z value of sample mean | | | | | | zc | =SQRT(40)*STANDARDIZE(3,5,2) | -6.32456 | | | | formula: zc=sqrt(n)*z = STANDARDIZE(x,mean,sd) | | | | | | | | | | | | | CR5) Determine the critical value of z-statistic (and/or Pcr) | | | | zcr = (=normsinv(pcr)) since 2-tailed test, use a/2 for pcr | | | | =NORMSINV(0.05/2) | | | -1.95996 | | | | | | | | | | | | NOTE: Since we are testing at either left or right, use zcr= abs= 1.96 | | | C5) compare the calculated valuesof test statistics against the corresponding | | | critical values: Decision: since zc (-6.32) > zcr (-1.96), the Ho is rejected. | | | This means that on the average life span of the battery is not equal to 5 years. |

OUTPUT PROBLEM 6 COMPARING MEANS OF DIFFERENT GROUPS | | | (2 insependent groups, n< 30), independent t-test | | | | | | | | | The average time for people to vote using the old procedure in a precint is 55 minutes. Using computers, | a random sample of 20 voters was found to have an average time to vote in 30 mins. with a | standard deviation of 1.5 mins. Determine if the average time using in computer is significantly | lower than average time using the old procedure. Use æ = 0.05 | | | | | | | | Solution: | | | | | | | 1) Write Ho and Ha | | | | | | | Ho: µ ≤ ẍ | Ha: µ old > x-bar | | | 2) Use α = 0.05 (1-tailed) | | | | | 3) Use sample t-test | | | | | | 4.) Compute the t-value for the sample mean | | | tc = sqrt(n)*standardize(x,mean,sd) | | | | = sqrt(20)*standardize(30,55,1.5) = | -74.5356 | | | 5) Determine critical t-value, (df = n-1), 1-tailed test | | tcr=tinv(0.05,19) = | 2.093024 | | syn: =tinv(α,df) | 6) Decision: | | | | | | Compare tc against tcr | | | | | Since tc (-74.5) > (-2.093), reject Ho | | | | this means that the time of voting using computers is | | much lower than the time using the old method. | | |

OUTPUT PROBLEM 7

Anova: Single Factor | | | | | | | | | | | | SUMMARY | | | | | | Groups | Count | Sum | Average | Variance | | | Column 1 | 5 | 66900 | 13380 | 11922000 | | | Column 2 | 5 | 67000 | 13400 | 1925000 | | | Column 3 | 5 | 71599 | 14319.8 | 2097710 | | | | | | | | | | | | | | | | | ANOVA | | | | | | | Source of Variation | SS | df | MS | F | P-value | F crit | Between Groups | 2882760 | 2 | 1441380 | 0.271196 | 0.767018 | 3.885294 | Within Groups | 63778841 | 12 | 5314903 | | | | | | | | | | | Total | 66661601 | 14 | | | | |

PROBLEM OUTPUT 8 SIMPLE LINEAR REGRESSION | | | | | | Problem 8 | | | | DATA | | | | Given the enrollment data in a new course | Years | Pupils | | | offering in school A, predict the number | | 1 | 75 | given | | of pupils in the next five years using | | 2 | 90 | given | | linear regression using | | | 3 | 96 | given | | | | | | | 4 | 135 | given | | a. Scatterplot diagram | | | 5 | 150 | given | | b. Regression function in | | | 6 | 168 | =(19.5*F9+50.7) | data analysis | | | | 7 | 187 | =(19.5*F10+50.7) | | | | | | 8 | 207 | =(19.5*F11+50.7) | | | | | | 9 | 226 | =(19.5*F12+50.7) | | | | | | 10 | 246 | =(19.5*F13+50.7) |

SUMMARY OUTPUT | | | | | | | | | | | | | | | | | Regression Statistics | | | | | | | | Multiple R | 0.969825 | | | | | | | | R Square | 0.940561 | | | | | | | | Adjusted R Square | 0.920748 | | | | | | | | Standard Error | 8.94986 | | | | | | | | Observations | 5 | | | | | | | | | | | | | | | | | ANOVA | | | | | | | | | | df | SS | MS | F | Significance F | | | | Regression | 1 | 3802.5 | 3802.5 | 47.47191 | 0.006264 | | | | Residual | 3 | 240.3 | 80.1 | | | | | | Total | 4 | 4042.8 | | | | | | | | | | | | | | | | | Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | Lower 95.0% | Upper 95.0% | Intercept | 50.7 | 9.386693 | 5.401263 | 0.01244 | 20.82735 | 80.57265 | 20.82735 | 80.57265 | X Variable 1 | 19.5 | 2.830194 | 6.889986 | 0.006264 | 10.49306 | 28.50694 | 10.49306 | 28.50694 |

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