Note: For all questions in this assignment that require you to use a two-sample t-test. always use the non-pooled procedure (that is, the procedure that assumes population standard deviations are not equal]. Further, when using the non-pooled procedure, use the following formula to calculate the degrees of freedom 2 2 2 5_1+5_2 n1 71: df= 2 (..:.)(i:)z+(..:.)(%)z' Note: if your t-table does not include the correct df value, please round down to the next available df value. STAT 15] Assignment 5 To'rol: 110 marks Part A 1. Suppose X1 is normally distributed with a mean of 3 and a standard deviation of 1 and suppose X2 is normally distributed with a mean of 5 and a standard deviation of 2. [a] For independent samples of sizes 8 and 10, respectively, nd the mean and standard deviation of f1 f2. (3 marks] (b) Would your answers in (a) change if X1 and/ or X 2 were not normally distributed? Explain your answer. [2 marks] [c] Explain why J71 )72 is normally distributed, despite the fact that both in and 112 are small. [2 marks) [d] Suppose two independent samples are randomly obtained from populations 1 and 2, with sample sizes 111 = 8 and 112 = 10. Compute the probability that either sample average is at least 3 greater than the other. [6 marks] 2. A study compared oral diversity between croplands and wetlands. Independent random samples of 126 cropland regions and 98 wetland regions were obtained in the Southern Great Plains. The cropland regions contained an average of 14.06 native oral species, with a standard deviation of 4.83; the wetland regions had an average of 15.36 and a standard deviation of 4.95. STAT 15] Assignment 5 To'rol: l 10 marks Part A 1. Suppose X1 is normally distributed with a mean of 3 and a standard deviation of 1 and 2. suppose X 2 is normally distributed with a mean of 5 and a standard deviation of 2. (a) For independent samples of sizes 8 and 10, respectively, nd the mean and standard deviation of )71 Y2. (3 marks] (b) Would your answers in (a) change if X1 and / or X 2 were not normally distributed? Explain your answer. [2 marks] [c] Explain why )71 )72 is normally distributed, despite the fact that both 111 and 112 are small. (2 marks) [d] Suppose two independent samples are randomly obtained from populations 1 and 2, with sample sizes 711 = 8 and 112 = 10. Compute the probability that either sample average is at least 3 greater than the other. [6 marks] A study compared oral diversity between croplands and wetlands. Independent random samples of 126 cropland regions and 98 wetland regions were obtained in the Southern Great Plains. The cropland regions contained an average of 14.06 native oral species, with a standard deviation of 4.83; the wetland regions had an average of 15.36 and a standard deviation of 4.95. (a) Test if the mean number of native oral species is lower in cropland regions. Use the 1% significance level. (8 marks) (b) Obtain a two-sided 99% condence interval for the difference in mean number of oral species between cropland and wetland regions. [4 marks] The Air Quality Index (AQI) is used to measure air quality near roadways and industrial regions in large cities. Governments report AQI values for common pollutants to keep the public informed about air quality. AQI values of 0-50 are regarded as good, 51-100 as moderate, 101-150 as unhealthy for sensitive individuals, 151-200 as unhealthy, 201-300 as very unhealthy, and beyond 300 as hazardous. The following table gives AQI values for S02 [Sulphur Dioxide] and C0 (Carbon Monoxide] from a random sample of 20 locations in California, between the years 2000 and 2010. STAT 151 Assignment 5 Total: 1 10 marks Location SO2 AQI CO AQI AQI Diff (SO2-CO) 1 1 3 -2 2 4 6 -2 3 w 6 -3 4 3 2 1 15 11 31 -20 6 0 8 -8 7 1 6 -5 8 27 5 22 9 7 16 -9 10 3 8 -5 11 0 7 -7 12 4 5 - 1 13 7 2 5 14 21 44 -23 15 3 5 -2 16 3 8 -5 17 4 5 -1 18 0 N - 2 19 5 -4 20 9 12 -3 Summary Statistics x1 = 5.6; s = 7.03 X2 = 9.3; s = 10.42 Xa = -3.7; s = 8.84 (a) Why are these data best regarded as a paired sample? (2 marks) (b) Do the data provide evidence that California's S02 and CO AQI levels differ, on average? Test at the 10% level of significance (8 marks) (c) Construct and interpret a 95% confidence interval for the mean difference in SO2 and CO AQI levels for California. (6 marks) (d) Does the interval from (c) lead to a conclusion that agrees with the test from (b)? Explain why or why not. (2 marks) (e) The following plots display the sample distribution of paired differences. Do the plots provide any clear indication against the assumption of a normally distributed population? Explain. (2 marks) Paired SO2-CO AQI Values Paired SO2-CO AQI Values Paired $02-CO AQI Values 10 D 10 Paired Differences (502-CO Paired Differences ($02-co) frequency -10 -10 -20 -10 20 Paired Differences (SO2-CO) norm quanties 3STAT 15] Assignment 5 Total: no marks 4. According to an online source, 79.8% of students are right-handed in Alberta secondary schools, 7.9% are left-handed, and 12.3% are ambidextrous (source: Census at School Canada, 2018-2019). A researcher wishes to test if these percentages seem reasonable among the general population of Albertans, so she obtains a random sample of 200 Albertans, from which she obtains the following frequency distribution: Handedness Right-Handed Left-Handed Ambidextrous Frequency 147 35 18 Does the above give evidence to suggest the proportions of right-handed, left-handed, and/ or ambidextrous Albertans differ from the proportions stated in the online source? Test at the 1% signicance level. [8 marks] 5. A plumbing company is interested in whether the reasons for service calls tend to differ between weekdays and weekends. The following table summarizes a random sample of 400 service calls, giving both the reason why the call was made and whether or not it was the weekend; do the data provide evidence of an association between the two variables? Test at the 5% signicance level. (8 marks) Weekend Reason for Call No Estimation 4-4 Re i air 130 Installation 120 Other 16 Total 3 10