Example 9.3: Let X1, X2, ..., Xn be a random sample from a population mean and variance o². Show that the sample variance, S² - Σ(x₁ - x)²
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- 4. Review the variance function for specific distributions. For Bernoulli, Poisson, Gamma, and Inverse Gaussian, what does the variance function say about our ability to predict the range of possible values? (This may be a little bit challenging; give it your best guess.)The aim of a study is to test the ratio of variances in the weight of two groups of rats being under a certain drugs A ( group 1) and B ( group 2) . A random sample of 16 rats was selected from group land 13 rats from group 2, Then their weight were recorded. The results are shown in the following table n Mean Median sd group 1 16 3.5 3.0 1.05 group 2 13 2.20 2.0 1.15 F15,12,0.05 =0.404 F19,16,0.025 = 0.386| F16,10,0.025 =0.335 F16,10,0.05 =0.401| F15,12,0.1 = 0.496 | F19,16,0.05 =0.451 F10,16,0.025 =0.286 F12,15,0.05 =0.382 F16,19,0.025 =0.371 F10,16,0.05 =0.354 F12,15,0.1=0.475 F16,19,0.05 = 0.437 Construct a 95% confidence interval for the ratio of the variances. [ Hint: because of the rounding select the closest interval ] O a. None of these O b. [ 2.488, 0.238 ] O c. [ 2.160 , 0.309 ] O d. [2.063 , 0.318]Experiment 3: Fifteen female CSCC students and 10 male CSCC students are randomly selected to determine if the mean time studying for females is significantly more than the mean study time for males. It can be assumed that all study times are normally distributed. Are these samples independent or dependent? Find the null and alternative hypothesis. H0: Ha:
- Let X denote the sample average for a random sample with mean uu and variance 02. Consider the following estimator for u: W=n+2/n+1°X Is W as consistent estimator of u? a. Yes b. No c. It dependsThe variance in a production process is an important measure of the quality of the process. A large variance often signals an opportunity for improvement in the process by finding ways to reduce the process variance. Jelly Belly Candy Company is testing two machines that use different technologies to fill three pound bags of jelly beans. The following tables contain a sample of data on the weights of bags (in pounds) filled by each machine. Machine 1 Machine 2 Ho: 01 2 Ho: 01 State the null and alternative hypotheses. 2 2 Ha: 01 Ha: 01 2.95 3.45 3.50 3.75 3.48 3.26 3.33 2 Ha: 01 ≤0₂ 3.16 3.20 3.22 3.38 3.90 3.36 3.25 2 3.20 2 3.12 Ho: 01 Conduct a statistical test to determine whether there is a significant difference between the variances in the bag weights for two machines. Use a 0.05 level of significance. What is your conclusion? = 2 Ho: 01 #0 2 02 = 2 2 02 02 2 2 2 ≤02 3.20 2 2 H₂: 01 > 02 3.28 3.22 2.98 3.45 3.70 3.34 3.18 3.35 3.22 3.30 3.34 3.28 3.29 3.25 3.30 3.27 3.38 3.34…1 Select one: ANB AUB NONE The variance of the following data equals: 0. 1 3 4 f (X) 1/ 3. 6/ Select one: NONE 4/3 6) 3.
- Compute mean, variance, and standard deviation1. Six batteries taken at random from a large lot were subjected to life tests with the following results: 19, 22, 18, 16, 25, and 20 hours. Find the probability that the sample mean of x= 20 hours, as observed here, or one less than 20 hours could arise if the true mean of the lot of batteries is 21.5 hours, assuming a known variance of o = 10 hours, for the individual batteries in the lot. %3DChild Health and Development Studies (CHDS) has been collecting data about expectant mothers in Oakland, CA since 1959. One of the measurements taken by CHDS is the weight increase (in pounds) for expectant mothers in the second trimester. In a fictitious study, suppose that CHDS finds the average weight increase in the second trimester is 14 pounds. Suppose also that, in 2015, a random sample of 40 expectant mothers have mean weight increase of 15.8 pounds in the second trimester, with a standard deviation of 6.2 pounds. A hypothesis test is done to see if there is evidence that weight increase in the second trimester is greater than 14 pounds. Find the p-value for the hypothesis test. The p-value should be rounded to 4 decimal places.
- A transect is an archaeological study area that is mile wide and 1 mile long. A site in a transect is the location of a significant archaeological find. Let x represent the number of sites per transect. In a section of Chaco Canyon, a large number of transects showed that x has a population variance . In a different section of Chaco Canyon, a random sample of 23 transects gave a sample variance for the number of sites per transect. Use an to test the claim that the variance in the new section is greater than 36.4. What is the level of significance? 90% 1% 5% 95% 10%A transect is an archaeological study area that is 1/5 mile wide and 1 mile long. A site in a transect is the location of a significant archaeological find. Let x represent the number of sites per transect. In a section of Chaco Canyon, a large number of transects showed that x has a population variance ?2 = 42.3. In a different section of Chaco Canyon, a random sample of 19 transects gave a sample variance s2 = 46.9 for the number of sites per transect. Use a 5% level of significance to test the claim that the variance in the new section is greater than 42.3. Find a 95% confidence interval for the population variance. (b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.)- 19.96 What are the degrees of freedom? -18 (c) Find or estimate the P-value of the sample test statistic. P-value > 0.1000.050 < P-value < 0.100 0.025 < P-value < 0.0500.010 < P-value < 0.0250.005 < P-value < 0.010P-value < 0.005…An experiment where a patient's blood pressure is measured before a treatment (call it X;) and after the treatment (call it Y;). The difference in the blood pressue of each patient, Z; = Y; – X; is also recorded. The sample size n of patients is large. Suppose the true population mean of the patient's blood pressure before treatment is 41 and after treatment is 42, both unknown. Suppose the true population variance of the patient's blood pressue before and after treatment are the same, but unknown. Find an approximate 1-a confidence interval for 12 - 41- 2.
