A sample of certain observations selected from a normally distributed population produced a sample variance of 38. Construct a 98% confidence interval for the population variance for each of the following cases and comment on the intervals as the sample size increases. (а) п — 12 (b) п — = 16 (с) п — 20 (а) п — 12
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- 4. Two populations are surveyed with simple random samples. A sample of size n is used for population I, which has a population standard deviation ; a sample of size 2 2n is used for population II, which has a population standard deviation σ-= 2σι. Ignoring finite population corrections, in which of the two samples would you expect the estimate of the population mean to be more accurate?Let X11 X12X1, and X2₂1, X22, X2na be two independent random samples of size ni and n₂ from two normal populations N(₁, of) and N(2, 2) respectively. (d) Derive a (1-a) x 100% confidence interval for (₁-2) when both samples are small but of and o are known. (e) Discuss how the confidence interval in part (c) will change if both the samples are large.6. A sample of certain observations selected from a normally distributed population produced a sample variance of 38. Construct a 98% confidence interval for the population variance for each of the following cases and comment on the intervals as the sample size increases. (а) п — 12 (b) п — 16 (c) n = 20 (а) п — 12
- Given a normally distributed population with a mean of 80 and a variance of 100, we know that the distribution of sample means computed from samples of size 25 from that population will have a mean of and a standard error of А. 50, 2 В. 80,10 С. 60, 0.5 D. 80, 2D. (U.33165, 0.40264) 28. Bob draws a simple random sample from an original population where the standard deviation is known to be a= 12. The sample size is n = 75 and the sample mean turns out to be T= 43. Find the 99% confidence interval for the population mean u. A. (33.431, 40.569) В. (35.431, 42.569) С. (37.431, 44.569) D. (39.431, 46.569)2. (20) Are Men and Women Equal Talkers? A headline in USA Today proclaimed that “Men, women are equal talker.” The headline referred to a study of the numbers of words that samples of men and women spoke in a day. Data given below are the results from the study. Assume that thedata are from two independent normal populations. Number of Words Spoken in a DayMen Womenn1 = 121 n2 = 101x1 = 18000 x2 = 17000s1 = 2000 s2 = 2500 Use a 10% level of significance, do a hypothesis test to determine whether or not the population variances are the same. (Hint: This is a two detailed F test. You need the conclusion here to make a choice between a pooled or a non-pooled t test for part (b).) (10) Based on your conclusion in part (a), use a 1% level of significance to test the claim that men and women speak the same mean number of words in a day by using critical value approach, which means you need to do a two detailed t test to test whether or not men and women…
- X. is found to be 19.1, and the A simple random sample of sizen is drawn from a population that is normally distributed. The sample mean, sample standard deviation, s, is found to be 4.9. (a) Construct a 96% confidence interval about u if the sample size, n, is 39. (b) Construct a 96% confidence interval about u if the sample size, n, is 68. How does increasing the sample size affect the margin of error, E? (c) Construct a 98% confidence interval about u if the sample size, n, is 39. How does increasing the level of confidence affect the size of the margin of error, E? (d) If the sample size is 14, what conditions must be satisfied to compute the confidence interval? (a) Construct a 96% confidence interval about u if the sample size, n, is 39. Lower bound: Upper bound: (Round to two decimal places as needed.) (b) Construct a 96% confidence interval about u if the sample size, n, is 68. Lower bound: ; Upper bound: (Round to two decimal places as needed.) How does increasing the sample…You have taken a random sample of size 18 from a normal population that has a population mean of -10 and a population standard deviation of -3. Your sample, which is Sample 1 in the table below, has a mean of 7-s. (In the table, Sample 1 is written "S1", Sample 2 is written "S2", etc.) 8 (4) Based on Sample 1, graph the 80% and 95% confidence intervals for the population mean. Use 122 for the critical value for the 80% confidence interval, and use 1.960 for the critical value for the confidence interval. (If necessary, consult a list of formulas • Enter the lower and upper limits on the graphs to show each confidence interval. Write your answers with one decimal place. • For the points (and), enter the population mean, μ-100. STA 970 970 80% confidence interval COM 100 95% confidence interval 80% 80% 95% 95% lower upper lower upper Smit limit limit limit 101.3 9.0 1018 101 3 990 1018 100.5 98.2 1010 1004 981 1009 100.7 98.4 101.2 100.5 98.2 100.0 100.7 100.7 984 1012 1012 98.9 101.7 $2…Two groups of children, one belonging-to the professional class, 125 in number, and the other belonging to the labour class, 124 in number are compared and the following results are obtained : Poor children Well-to-do children per cent (Labour class) per cent (Professional class) Below normal weight 55 13 Above normal weight 11 48 Find the coefficient of association between the weight of the children and their social status. Also calculate the standard error of this coefficient of association and set 95% fiducial limits of this population coefficient.
- 4b.1A random sample of 12 pea tins from a canning factory's production is selected on a given day, and their contents are weighed. For the sample, the weight's mean and standard deviation are, respectively, 301-8 g and 1-8 g. The mean weight of peas in tins produced by the factory on the relevant day should have 99% confidence limits.The mean and standard deviation of the contents for this sample are 302-1 gm and 1-6 gm, respectively, the day after another random sample of 12 tins is obtained. AssumingA 95% confidence interval demonstrates that the variances of the weights are the same on the two days interval includes 0 for the mean weight difference between the two days.NextNow suppose that the samples from both days come from the same population. Using both samples, calculate a 99% confidence interval for the mean weight of tins in that population.(b) The following data is obtained from a sample. n- 50 Ex= 2685 Er- 144346 (i) Find unbiased estimates for the population mean and variance. (i) Calculate a 99% confidence interval for the mean u. In a public opinion poll, 250 randomly chosen voters were interviewed and 90 of them said they would vote for 'PPM' party in the next general elections. The leader of the party states that 40% of all the voters support his party. Test at 5% level whether the leader is overstating his support21 - The average weekly income of 50 students randomly selected from a class is 250 TL with a variance of 16, a weekly average working time of 15 hours and a variance of 16. The average weight of the students was 60 kg and the variance was 9, and the average of their height was 160 cm and the variance was 25. Based on this information, which of the following is true of the variable with the greatest variation between observations? a) Income variable B) Income and working time variable NS) Height variable D) Working time variable TO) Weight variable