MBA503 MidTerm

Descriptive Statistics 1) Egon Pearson, the son of Karl Pearson, was a prominent statistician known for his contributions to statistical theory and hypothesis testing. He worked alongside his father and developed the concept of the Pearson correlation coefficient. Egon Pearson lived from 1895 to 1980. Carl Friedrich Gauss, commonly known as Carl F. Gauss, was a German mathematician, physicist, and astronomer. He made significant contributions to various fields, including number theory, algebra, statistics, and geodesy. Gauss is often referred to as the "Prince of Mathematicians" and his work had a profound impact on the development of modern mathematics. He lived from 1777 to 1855 and his contributions continue to be studied and celebrated to this day. William Edwards Deming was an American statistician who focused on quality control and management. He emphasized the importance of statistical methods in improving industrial processes and promoting quality. Deming's work played a significant role in the development of Total Quality Management (TQM). He lived from 1900 to 1993 and his contributions continue to influence the field of statistics and quality management today. 2) (a) To visualize the distribution of missed days for all workers, we can create a histogram or a box plot. This will help us determine if the distribution is symmetric, skewed left, or skewed right. (b) Switching from the mean to the median as the trigger point for the union absenteeism penalty may have an effect on the trigger point. The median is less affected by extreme values or outliers compared to the mean, which is influenced by all values. This means that if there are a few workers with a significantly higher number of missed days, the median may be less affected by their presence. (c) The union's position on the company's proposed switch would depend on various factors, such as the distribution of missed days and the union's priorities. Generally, the union may support the switch to the median if there are concerns about outliers unfairly affecting the penalty. However, if the majority of workers have a low number of missed days and outliers are not a significant issue, the union may prefer to keep the mean as the trigger point. 3) Column1 Mean 724.6666667 Median 720 Mode 730 Standard Deviation 114.2813628 b) Although the mean and median are relatively close, the mode is not close to the median and skewedness. Therefore, the measures of central tendency do not agree.

c) Standard Deviation 114.28 d). Data Zstandardized 500 -1.9659608 560 -1.4409345 570 -1.3534302 600 -1.090917 620 -0.9159083 620 -0.9159083 650 -0.6533952 660 -0.5658908 670 -0.4783864 690 -0.3033777 690 -0.3033777 700 -0.2158733 700 -0.2158733 710 -0.1283689 720 -0.0408645 720 -0.0408645 730 0.0466398 3 730 0.0466398 3 730 0.0466398 3 730 0.0466398 3 740 0.1341442 1 740 0.1341442 1 760 0.3091529 6 800 0.6591704 6 820 0.8341792 1 840 1.0091879 6 850 1.0966923 3 930 1.7967273

b) The higher the GDP per Capita, the lower the birth rate. This is linear. Chi-Square tests – cross tab- not included. ((Observed – expected)^2)/Expected (O-E)^2/E 18-27 28-37 38-47 48-57 58-67 Coupon 45.92509835 0.5727 120.27 73.467 65.359 No Coupon 19.77306006 0.2466 51.783 31.631 28.14 x 437.1683828 df = (#rows-1)(column-1) 4 p 2.58043E-93 X is the summation of all the ((Observed – expected)^2)/Expected values. Df is the degree of freedom. Number of rows minus 1 times number of columns minus 1. p-value is CHISQT.TEST(X,df). In this situation, since the p-value < 0.001, the results are statistically significant. It’s an extremely small value indicating strong evidence against the null hypothesis.

7) a) The graph is more helpful in describing the salad sales by Noodles & Company. This is because it shows trends and it is more visual. b) The most sales were made in May while the least sales were made in December. Sampling Distribution and Estimators 8) a) 270.0000 n 10 N n/N FPCF C.L. (e.g., 95%) 95% Alpha 0.0500 Alpha/2 0.0250 Sample Std Dev ( s ) 20.0000 Student's t 2.2622 Standard Error 6.3246 FPCF Margin of Error 14.3071 Confidence Interval Lower 255.6929 Upper 284.3071 b) 270.0000 n 20 N n/N FPCF C.L. (e.g., 95%) 95% Alpha 0.0500 Alpha/2 0.0250 Sample Std Dev ( s ) 20.0000 Sample Mean (´ x μ Sample Mean (´ x

a) ER patient: - Type I error: Rejecting the null hypothesis (no heart attack) when it is actually true (chest pain due to muscle pain). - Type II error: Failing to reject the null hypothesis (no heart attack) when it is actually false (there is a heart attack). - Costs: Type I error could lead to unnecessary medical interventions and use of scarce hospital resources. Type II error could delay necessary treatment for a heart attack. b) British Air flight: - Type I error: Declaring an emergency and landing immediately when there is enough fuel to stay aloft for 15 more minutes. - Type II error: Failing to declare an emergency and continuing to hold, risking fuel depletion. - Costs: Type I error could lead to an investigation and potential endangerment of other flights. Type II error could result in fuel exhaustion and emergency landing. c) Printer ink: - Type I error: Going to Staples for a new ink cartridge when there is actually enough ink. - Type II error: Not going to Staples for a new ink cartridge when there is not enough ink. - Costs: Type I error could waste time and effort going to the store unnecessarily. Type II error could result in not having enough ink to print the report. 10) Null Hypothesis (H0): The true mean is equal to or greater than the specification (μ ≥ 18) Alternative Hypothesis (H1): The true mean is smaller than the specification (μ < 18) 18.00 17.78 0.41 18 0.05 -2.2765 Critical Value = -1.7396 Since t-value is less than the critical t-value, reject the null hypothesis and conclude that the true mean is smaller than the specification. b) Yes, the conclusion of a hypothesis test can be sensitive to the choice of the level of significance. The level of significance, also known as alpha (α), determines the probability of making a Type I ´ x = ¿ s = ¿ n = α = t calc = ¿

error, which is rejecting the null hypothesis when it is actually true. Choosing a higher level of significance increases the chances of rejecting the null hypothesis, while choosing a lower level of significance decreases the chances of rejecting the null hypothesis. c) p -value = 0.0180 The p-value is less than the level of significance so reject the null hypothesis.