Q3W2_Chi-Square Distribution (1Q3W2_Chi-Square Distribution (1).ppt).ppt

Editor's Notes

• #3: Previously, we have made inferences about population means and proportions. We can make similar inferences about population variances.
• #4: In these cases, we are dealing with population variances (not means or proportions)
• #9: In studying population variances, the following quantity is important
• #10: Is there a relationship between student gender and course choice?"; and the goodness-of-fit test, which asks something like "How well does the coin in my hand match a theoretically fair coin?"
• #11: Notice in the graph below how the confidence level, the significance level and the critical values ��2XR2​ and ��2XL2​ are shown. The R and L subindexes denote the sides right and leftsince we are looking at a two tailed distribution, and notice the significance level is still evenly divided in halves (one for each side) even if the distribution is not symmetrical.
• #12: Notice in the graph below how the confidence level, the significance level and the critical values ��2XR2​ and ��2XL2​ are shown. The R and L subindexes denote the sides right and leftsince we are looking at a two tailed distribution, and notice the significance level is still evenly divided in halves (one for each side) even if the distribution is not symmetrical.
• #13: Notice in the graph below how the confidence level, the significance level and the critical values ��2XR2​ and ��2XL2​ are shown. The R and L subindexes denote the sides right and leftsince we are looking at a two tailed distribution, and notice the significance level is still evenly divided in halves (one for each side) even if the distribution is not symmetrical.
• #14: From the chi square formula we can solve for the value of the variance (the square of the populations standard deviation). With this we can calculate the confidence interval for that variance too, similarly to how we have used the t-distribution and the margin of error before to calculate the confidence interval for a population mean. The confidence interval, in this case for the variance or standard deviation squared, is given by: In this last formula, the chi square values in the denominators of each side refers to the chi square critical values defining the edges between the confidence level and the significance level on the distribution graph, and denoting that we have a two tailed significance level because we have a right side (��2XR2​) and a left side (��2XL2​) as shown below:
• #15: Is there a relationship between student gender and course choice?"; and the goodness-of-fit test, which asks something like "How well does the coin in my hand match a theoretically fair coin?"
• #20: Demographics are statistics that describe populations and their characteristics. Demographic analysis is the study of a population-based on factors such as age, race, and sex.
• #24: A=0.05 A/2=0.025 X right = 32.852 Xleft = 8.907
• #27: A=0.10 A/2=0.05 X right = 23.685 Xleft = 6.571
• #30: A=0.01 A/2=0.005 X right = 42.796 Xleft = 8.643
• #31: A=0.01 A/2=0.005 X right = 38.582 Xleft = 6.844 Variance = 1.4 and 8.02 Sd=1.2 and 2.83
• #32: X right = 32.852 Xleft = 8.907 X right = 32.852 Xleft = 8.907 X right = 32.852 Xleft = 8.907 X right = 32.852 Xleft = 8.907 X right = 32.852 Xleft = 8.907
• #33: We can see form the table that the two critical values are 4.404 and 23.337.
• #34: We can see form the table that the two critical values are 4.404 and 23.337.
• #39: The temperature readings of the ten thermostats are shown on the next slide.
• #46: The temperature readings of the ten thermostats are shown on the next slide.
• #52: A=0.01 A/2=0.005 X right = 38.582 Xleft = 6.844 Variance = 1.4 and 8.02 Sd=1.2 and 2.83
• #53: A=0.01 A/2=0.005 X right = 38.582 Xleft = 6.844 Variance = 1.4 and 8.02 Sd=1.2 and 2.83
• #54: A=0.05 A/2=0.025 X right = 32.852 Xleft = 8.907
• #55: A=0.05 A/2=0.025 X right = 32.852 Xleft = 8.907
• #56: A=0.05 A/2=0.025 X right = 32.852 Xleft = 8.907
• #58: A=0.05 A/2=0.025 X right = 32.852 Xleft = 8.907
• #59: herefore the right critical value is ��2XR2​ = 23.685 . Therefore the right critical value is ��2XL2​ = 6.571 . Therefore: 5.911<  �2  <21.306 5.911<σ2<21.306
• #60: Therefore the right critical value is ��2XR2​ = 66.766 . Therefore the right critical value is ��2XL2​ = 20.707. Therefore: 2.322<  �  <4.169 2.322<σ<4.169
• #61: Therefore the right critical value is ��2XR2​ = 66.766 . Therefore the right critical value is ��2XL2​ = 20.707. Therefore: 2.322<  �  <4.169 2.322<σ<4.169
• #62: Therefore the right critical value is ��2XR2​ = 66.766 . Therefore the right critical value is ��2XL2​ = 20.707. Therefore: 2.322<  �  <4.169 2.322<σ<4.169
• #63: Therefore the right critical value is ��2XR2​ = 66.766 . Therefore the right critical value is ��2XL2​ = 20.707. Therefore: 2.322<  �  <4.169 2.322<σ<4.169
• #64: Therefore the right critical value is ��2XR2​ = 66.766 . Therefore the right critical value is ��2XL2​ = 20.707. Therefore: 2.322<  �  <4.169 2.322<σ<4.169
• #65: For a sample size of   n=27�=27,   we will have   df=n−1=26��=�−1=26   degrees of freedom. For a 95% confidence interval, we have   α=0.05�=0.05,   which gives 2.5% of the area at each end of the chi-square distribution. We find values of   χ20.975=13.844�0.9752=13.844   and   χ20.025=41.923�0.0252=41.923.   Evaluating (n−1)s2χ2(�−1)�2�2, we obtain 21.297 and 64.492. This leads to the inequalities   21.297≤σ2≤64.49221.297≤�2≤64.492   for the variance, and taking square roots,   4.615≤σ≤8.0314.615≤�≤8.031   for the standard deviation.
• #66: First, we need to determine the two chi-square values with (n−1) = 9 degrees of freedom. Using the table in the back of the textbook, we see that they are: �=�1−�/2,�−12=�0.975,92=2.7 and �=��/2,�−12=�0.025,92=19.02 Now, it's just a matter of substituting in what we know into the formula for the confidence interval for the population variance. Doing so, we get: (9(4.2)19.02≤�2≤9(4.2)2.7) Simplifying, we get: (1.99≤�2≤14.0) We can be 95% confident that the variance of the weights of all of the packs of candy coming off of the factory line is between 1.99 and 14.0 grams-squared. Taking the square root of the confidence limits, we get the 95% confidence interval for the population standard deviation �: (1.41≤�≤3.74) That is, we can be 95% confident that the standard deviation of the weights of all of the packs of candy coming off of the factory line is between 1.41 and 3.74 grams.
• #67: 0.025 32.852 8.907 1.5<o<5.5 1.2<o<2.3
• #68: Df=9 S=5.31 S2=28.22 Right=16.919 Left=3.325 15.0 o276.3 3.87 0 8.73
• #69: Df=10 S2=1.3412 Right=18.307 Left=3.940 02=0.9826, 4.5655 0=.9913, 2.1367 Mean=94.0455 Cv=+-1.812 Sd=1.3412 Df=10 N=11 Ci=93.3128, 94.7782
• #70: The mean of the previous source of grain was 94.2. This lies in the middle of the confidence interval calculated for the mean of the new source of grain. There is therefore no evidence that the means differ. The sd of the previous source of grain was 2.5 and hence the variance is 6.25. This is above the upper limit of the confidence interval for the variance of the new source of grain. This suggests that the new source gives less variability Combining these two conclusions suggests that the new sources is preferable to the previous source.