Download presentation

Presentation is loading. Please wait.

Published byJayden Carlson Modified over 3 years ago

1
Lesson 14 - 1 Test for Goodness of Fit One-Way Tables

2
Knowledge Objectives Describe the situation for which the chi-square test for goodness of fit is appropriate Define the χ 2 statistic, and identify the number of degrees of freedom it is based on, for the χ 2 goodness of fit test List the conditions that need to be satisfied in order to conduct a test χ 2 for goodness of fit Identify three main properties of the chi-square density curve

3
Construction Objectives Conduct a χ 2 test for goodness of fit. Use technology to conduct a χ 2 test for goodness of fit. If a χ 2 statistic turns out to be significant, discuss how to determine which observations contribute the most to the total value.

4
Vocabulary Goodness-of-fit test – an inferential procedure used to determine whether a categorical frequency distribution follows a claimed distribution. Expected counts – probability of an outcome times the sample size for k mutually exclusive outcomes One-way table – a table of k mutually exclusive observed values Cells – one item in the one-way table

5
Chi-Square Distributions

6
Chi-Square Distribution Total area under a chi-square curve is equal to 1 It is not symmetric, it is skewed right The shape of the chi-square distribution depends on the degrees of freedom (just like t-distribution) As the number of degrees of freedom increases, the chi-square distribution becomes more nearly symmetric The values of χ² are nonnegative; that is, values of χ² are always greater than or equal to zero (0); they increase to a peak and then asymptotically approach 0 Table D in the back of the book gives critical values

7
Chi-Square Test for Goodness of Fit

8
Conditions Goodness-of-fit test: Independent SRSs All expected counts are greater than or equal to 1 (all E i ≥ 1) No more than 20% of expected counts are less than 5 Remember it is the expected counts, not the observed that are critical conditions

9
Critical Region Reject null hypothesis, if P-value < α χ 2 0 > χ 2 α, k-1 P-Value is the area highlighted χ2αχ2α P-value = P(χ 2 0 ) Goodness-of-Fit Test where O i is observed count for ith category and E i is the expected count for the ith category ( O i – E i ) 2 Test Statistic: χ 2 0 = ------------- E i Σ (Right-Tailed)

10
Things to Avoid

11
Example 1 Are you more likely to have a motor vehicle collision when using a cell phone? A study of 699 drivers who were using a cell-phone when they were involved in a collision examined this question. These drivers made 26,798 cell phone calls during a 14 month study period. Each of the 699 collisions was classified in various ways. Are accidents equally likely to occur on any day of the week? SunMonTueWedThuFriSat 2013312615913611312

12
Example 1 – Graphical Analysis Are accidents equally likely to occur on any day of the week? SunMonTueWedThuFriSat 2013312615913611312

13
Example 1 – Chi-Square Analysis Are accidents equally likely to occur on any day of the week? Hypotheses: Conditions: H 0 : Motor vehicle accidents involving cell phones are equally likely to occur everyday of the week H a : Motor vehicle accidents involving cell phones will vary everyday of the week (not all the same) Expected counts (everyday) = 699/7 = 99.857 1)All expected counts > 0 2)All expected counts > 5

14
Example 1 – Chi-Square Analysis Are accidents equally likely to occur on any day of the week? Calculations: Interpretation: ItemSunMonTueWedThuFriSat Observed2013312615913611312 Expected99.86 χ²63.8611.006.8435.0313.081.7377.30 (Obs – Exp)² χ² = ∑ ----------------- Exp χ² = ∑ (63.86 + 11 + 6.84 + 35.03 + 13.08 + 1.73 + 77.3) = 208.84 Since our χ² value is much greater than the critical value (208 > 24.1), we would reject H0 and conclude that the accidents are not equally likely each day of the week. χ² (n-1,p-value) = χ² (6, 0.0005) = 24.1

15
Example 2 YellowOrangeRedGreenBrownBlueTotals Sample 1668838595396400 Sample 21094169755 Peanut0.150.230.120.150.120.231 Plain0.140.20.130.160.130.241 K = 6 classes (different colors) CS(5,.1)CS(5,.05)CS(5,.025)CS(5,.01) 9.23611.07112.83315.086

16
Example 2 (sample 1) H 0 : H 1 : Test Statistic Critical Value: Conclusion: YellowOrangeRedGreenBrownBlueTotals Observed668838595396400 Expected609248604892400 Chi-value0.60.1742.6320.0170.5210.1744.118 ( O i – E i ) 2 Test Statistic: χ 2 0 = ------------- E i Σ The big bag came from Peanut M&Ms The big bag did not come from Peanut M&Ms All critical values are bigger than 9! FTR H 0, not sufficient evidence to conclude bag is not peanut M&M’s

17
Example 2 (sample 2) H 0 : H 1 : Test Statistic Critical Value: Conclusion: YellowOrangeRedGreenBrownBlueTotals Observed1094169755 Expected8.2512.656.68.256.612.65400 Chi-value0.3711.0531.0247.2800.8732.52413.125 ( O i – E i ) 2 Test Statistic: χ 2 0 = ------------- E i Σ The snack bag came from Peanut M&Ms The snack bag did not come from Peanut M&Ms All critical values are less than 13, except for α = 0.01 ! Rej H 0, sufficient evidence to conclude bag is not peanut M&M’s

18
TI & Chi-Square (One-Way) Enter Observed values in L1 Enter Expected values in L2 Enter L3 by L3 = (L1 – L2)^2/L2 Use sum function under the LIST menu to find the sum of L3. This is the value of the χ² test statistic Largest values in L3 are the observations that are the largest contributors to the total value

19
Summary and Homework Summary –Goodness-of-fit tests apply to situations where there are a series of independent trials, and each trial has 3 or more possible outcomes –The test statistic to be used combines all of the outcomes and all of the expected counts –The test statistic has approximately a chi-square distribution –Calculator is a tool for one-way tables not a crutch! Homework –pg 846 14.1 – 14.6, 14.8

Similar presentations

OK

Comparing Three or More Means ANOVA (One-Way Analysis of Variance)

Comparing Three or More Means ANOVA (One-Way Analysis of Variance)

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on remote operated spy robot with camera Ppt on bond length chemistry Ppt on tsunami early warning system Ppt on chapter management of natural resources Ppt on recycling of waste paper Ppt on second law of thermodynamics states Ppt on road accidents videos Ppt on tamper resistant electrical outlets Ppt on personality development presentations Ppt on conceptual art pioneer