Brief Scenario: A Confidence Interval - Assignment Example

Answer the following 6 question CONFIDENCE INTERVAL A confidence interval is an estimation of where the population parameter lies. The populationparameter is the true MEAN. When we know the sample mean, we can determine the population mean within an interval. Here are some fighting politicians trying to fetch every vote they can gather. Refer to Lind, Chapter 8 (sampling) and 9 (confidence intervals). Very Brief Scenario: DQ: An upcoming homeowner’s association election is being held in a couple days. The Galloping Pollsters took a sample last week, and determined the following: (No significant math involved.) Bugsy Rabbit      49% +/- 2% Elmer Fuddruckers 50% +/- 2% Undecided          1% A) (1) What is another name for a point estimate? (2) What the point estimate for Bugsy? Answer: (1) Another name of point estimate is Sample Statistics. (2) Point estimate for Bugsy is 49%. B) What is the confidence interval for Elmer? Answer: Confidence interval for Elmer is 48% - 52%. C) What does it mean for Bugsy’s and Elmer’s confidence interval to overlap? Answer: This means there is no significant difference between the votes that Bugsy and Elmer are expected to get. Critical Thinking: D) Who has the greatest chance of winning if elections were held tonight (consider when the poll was taken)? Why? Answer: Elmer has greatest chance of winning if elections were held tonight. This is because, if one calculates the probability of Elmer getting more than 50% (which is nothing but area under normal curve towards right of z = 0 i.e. 50% of votes) this value (0.5) is higher than the same for Bugsy. E) Can these types of polls predict the winner? Why? Answer: No, this type of polls cannot predict winner, these tests can only give which contestant has higher chance of winning. This is because; statistical investigations can give only probabilities for an event and cannot predict an event with absolute certainty. In fact there is no test that can predict future with certainty. In this case the subjects are humans and not machines and circumstances may change dramatically from the time of survey to actual voting leading to unexpected results. 2. CONFIDENCE INTERVALS You may answer parts a-c and d-f in separate postings because this is a long activity. Attached is Jims data for the Small Business card that includes descriptive statistics for variables: D1   D6    B5 Attached is a Confidence Interval Tutorial & Calculator. Put the numbers in the green cells and the calculations are performed automatically. We will use Continuous and z-distribution for this activity. Pick ONE variable of your choice and identify it. D1 - Yrs_in_Bus.    D6 - #_Employees   B5 - Gross_Inc_$ a. Is the confidence interval an exact decision-making tool or as an estimation tool? Answer: Confidence interval is not an exact decision making tool in stead it is only an estimation tool. One can estimate a interval in which lies the exact population parameter. b. What is a Confidence Interval of your variable?     (Write with upper and lower values, Ex: 48.3 - 51.8) Answer: The 95% confidence interval for Yrs_in_Bus is 13.84 – 17.24 c. We have taken 78 samples using a 95% level of confidence. What do we know about the true population mean? Answer: This means if one takes 100 different samples with 78 data points in each and constructs 95% level confidence intervals the true population mean will lie in 95 of such confidence intervals and outside the remaining 5 intervals. Critical Thinking: Use these true mean values for the next questions. D1: True mean = 13.5 years D6: True mean = 4.9 employees B5: True mean = $2,000,000 d. Would it be reasonable to find the true mean within the Confidence Interval using a 95% confidence level? Answer: No, because the true mean value for each of these variables D1, D6 and B5 are outside the 95% confidence interval for these variables as derived from the given data. e. Would it be reasonable to find the true mean within the Confidence Interval using a 99% confidence level? Did the Confidence Interval increase or decrease in range from the 95% confidence level.  Answer: Confidence interval increases with increasing confidence interval. However, the true mean of variables D1 and D6 still does not lie in the 99% confidence interval. However, true mean of B5 lies in the 99% confidence interval of the given sample. e. Increase the sample size, n, to 1,000. Would it be reasonable to find the true mean within the Confidence Interval using a 99% confidence level? Did the Confidence Interval increase or decrease in range from sample size of 78?  Answer: Increasing the sample size from n = 78 to n = 1000 leads to narrowing or decrease of the confidence interval. However, the value of sample mean itself will change when the sample size is increased from n = 78 to n = 1000; leading to shift in the confidence interval for a sample with n = 1000 as compared to the same for a sample with n = 78. A larger sample is generally more representative of a true population than a smaller sample and therefore, it is reasonable to find true mean using a 99% confidence interval when a sample with n = 1000 is chosen. f. (1) Why is it better to have a narrow Confidence Interval? Answer: This is because a narrow confidence interval is more deterministic that means the value of true mean is then restricted in a small region.       (2) Would you increase or decrease the confidence level to decrease the Confidence Interval? Answer: Confidence level has to be decreased to decrease the confidence interval.       (3) Would you increase or decrease the sample size, n, to decrease the Confidence Interval? Answer: Sample size has to be increased to decrease the confidence interval. 3. SAMPLE SIZE DETERMINATION Attached is a sample size tutorial & calculator. Just input the values into the green boxes and the sample size automatically calculates. Definition:   Population size, N   Sample size, n We will use the Proportions calculator with Finite size population for this activity. a. (1) In general, what three factors (Lind, Ch. 9, p. 301, must be specified to calculate a sample size? Answer: Three factors that must be specified for calculating sample size are: i) Confidence level or corresponding z-value ii) Standard deviation or  iii) Acceptable error or e (3) How do each factor affect the calculation? For instance, if factor1 increases, will the sample size increase or decrease? Answer: Sample size increases with i) Increasing Confidence level or corresponding z-value ii) Increasing Standard deviation or  iii) Decreasing Acceptable error or e b. Per the Small Business Administration, there are approximately 5,700,000 small businesses with employees ending 2003. Use the sample size calculator to determine the sample size with a margin of error (E): (1) E=10%    Answer: (2) E=5%   Answer: Critical thinking: c. Jims team used a sample size = 78. (1) What is the approximate margin of error? (2) Suppose E=0.10 was required. What is your educated opinion - was the sample size of 78 sufficient? d. (1) If the actual collection of samples is less than desired, what would be the researchers main concern? Answer: If actual collection of samples is less then error will be more and this will be the main concern of the researchers. (2) What might be done in the planning stage to offset this dilemma? Answer: Collection of reasonably bigger sample should be planned to offset this dilemma. 4. HYPOTHESIS TESTING Refer to: Lind Ch 10, pp. 323-324 for definitions on tails and tail directions, and Lind Ch 10 and Ch 11 for the activity. DQ: (1) Given the research question, determine if the hypothesis is one- or two-tail. If a one-tail hypothesis, the direction left or right. (2) Explain you choices in a brief paragraph. Answer any two from each chapter for a total of four. A. Two-tail,         (H1: µ ≠ ) B. One-tail, Left   (H1: µ < ) C. One-tail, Right (H1: µ > ) (Note: H1 is the alternate hypothesis) Chapter 10         Research Questions a. # 5, p. 331    Is Crossets experience different from that claimed by the manufacturer? Answer: 1) This is a two-tail test. 2) In this case one will have to test whether the two experiences – one is the experience of Crosset’s and another is that claimed by the manufacturer. The Crosset’s experience can be on the either side of what is claimed by the manufacturer. b. #12, p. 335    Can we conclude that a larger proportion of students at your high school have jobs? Answer: 1) This is a one-tail, Right test. 2) In this case the sample proportion p will be on the right side of the population proportion . c. #18, p. 340    Can we conclude that the assembly time using the new method is faster? Answer: 1) This is a one-tail, Right test. 2) In this case the sample mean will be on the right side of the population proportion . Chapter 11         Research Questions d. # 3, p. 361    Can we conclude that basics using Gibbs brand have gained less weight? 1) This is a one-tail, Left test. 2) In this case the sample mean will be on the left side of the population proportion . e. #12, p. 365    Is there a significant difference in the proportions of single and married persons having an accident during a three-year period? Answer: 1) This is a two-tail test. 2) In this case the sample mean can be on either side of the population proportion . f. #18, p. 370    Is it reasonable to conclude that the mean weekly salary of nurses is higher than school teachers? Answer: 1) This is a one-tail, Right test. 2) In this case the sample mean will be on the right side of the population proportion . g. #22, p. 377   Has there been a decrease of crimes since the inauguration of the program? Answer: 1) This is a one-tail, Right test. 2) In this case the sample mean will be on the left side of the population proportion . 5. PROBABILITY VALUE Attached is the Instant Rewards Data modified for this Activity. Refer to Lind, Ch. 10, pp. 328-329. The text illustrates the five-step hypothesis testing method, which uses the comparison between the Critical Value and the Test Statistic to decide whether to reject the null hypothesis or not. While this method is still being used in industry, the computer calculated P-value provides better information towards making the decision. The P-value is the probability of rejecting a true hypothesis (the probability of making a mistake.) We compare the P-value to the level of significance, Alpha, to make a decision. If the P-value is less than Alpha, we reject the null hypothesis. This shows the data is statistically significant and generalizations can be made about the population from which the data came. In other words, the sample looks like the population. Caveat: We are looking at the same data that was given to Bea. Make the assumption for this DQ that at this point you do not know the data is invalid. DQ: a. It is preferable to have a low P-value (less than the level of significance, Alpha?) Answer: Yes it is preferable to have a low P-value. As P-value is the probability of making a mistake and smaller the probability of making a mistake, better it is. b. Would the P-values on Tables 3, 4, 5 results in rejecting the null hypothesis? Table 3: No. Table 4: Yes Table 5: Yes Critical thinking c. If the null hypothesis was rejected, does that show the data is statistically significant? Answer: Yes d. (1) Why might I want the data to statistically significant? Answer: This is because, statistically significant data implies the sample is very representative and conclusions regarding the population can be made from the sample. (2) If the data is not statistically significant, should I make generalizations about the population from which the data came? Answer: No, it cannot be done because the sample is not representative of the entire population. e. Interpret the finding for Table 5. Use the wording on page 2 in Step 1. Answer: The P-value 0.00. This is the probability of rejecting a true hypothesis (the probability of making a mistake.) As P-value (0.00) is less than -value (0.05, for 95% confidence level or 5% significance level); therefore, the Null Hypothesis is rejected and the Alternate Hypothesis is accepted. This means IR is significantly different from 5. 6. Provide one benchmark resource you CAN USE and how it applies to UWB (USA WORLDBANK)? Read More