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Statistical Analysis and Hypothesis Testing: A Comprehensive Study on Data Interpretation, Probabilities, and Confidence Intervals

A detailed study on statistical interpretation, probability calculations, and confidence intervals.

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    Statistical Analysis and Hypothesis Testing: A Comprehensive Study on Data Interpretation, Probabilities, and Confidence Intervals In this assignment you may use your book, your notes and your calculator. SHOW YOUR WORK WHERE STATED TO DO SO TO RECEIVE FULL CREDIT. Round all decimal answers to 4 decimal places. 1. The population of the City of Baltimore is 622,104 (as of July 1, 2013). Mayor Rawlings - Blake would like a survey of the residents of Baltimore to determine if they are satisfied with how Monday’s riots were handled. How could a survey be developed that is unbiased and addresses a representative sample of the population? Solution: A survey should be developed which is unbiased and sample should be taken using simple random sampling method. In simple random sampling, a sample is selected randomly from population of the City of Baltimore consisting of 622,104 people. Here, each member of the population has equal and independent chance of being selected in the sample. An unbiased survey should be conducted such that respondents answer the survey reflecting their true beliefs about Monday’s riots. 2. What is the difference between qualitative and quantitative data? Give an example of each. Solution: Categorical variables come under the category of Qualitative data. While data which can be measured on numeric scale that is they are represented as numbers comes under the category of quantitative data. For example, gender is qualitative data which can be categorized as male or a female . And, weight of person s is a quantitative data which can be represented as numeric like 30, 45, 65, 80, 110 Kg. Consider the data below for questions 3 -- 4.

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    3. What number best represents the 25th percentile? Solution: 90 4. Prove or disprove the existence of outliers. Solution: T here are no outliers present in the data, as there are no individual points on Box plot. 6. Determine the mean, median and mode for the data below. Which measure of central tendency best represents the data? Provide reasoning supporting your answer. Number of Days Absent per Student, Mr. Steven’s 6 th grade class 7 5 12 3 0 5 13 12 5 0 6 38 11 0 1 4 9 11 2 3 6 0 1 1 Solution: Mean = sum of all observations / total number of observations = 155/24 = 6.4583 For median, first of all arrange the data in the descending order which is shown below. 0 0 0 0 1 1 1 2 3 3 4 5 5 5

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    6 6 7 9 11 11 12 12 13 38 I know that, w hen n = even, M edian = average of (n/2)th and ((n/2) + 1)th observation Here, n = 24 , thus median = average of 24/2th and ((24/2)+1)th observation = average of 12 th and 13 th observation = (5+5)/2 = 10/2 = 5 Mode is the observation occurring maxi mum number of times. Here is 0 is occurring 4 times, which is the maximum frequency here. Thus, mode = 0 Here 38 is an outlier, And i n presence of outliers, Median is preferred over mode and mean. Thus, median is the best measure of central tendency. 7. Jason and Mary took an entrance exam with a certain company. Jason scored a 76%. His cohort’s average was 80% with a standard deviation of 5%. Mary also scored a 76%. Her cohort’s average was an 83% with a standard deviation of 8%. Determine a z - score for each person and compare their scores using z - scores. SHOW YOUR WORK Solution: For Jason, X = 0.76, average = 0.80 and standard deviation = 0.05. Z score = (X - mean)/s.d. = (0.76 - 0.80)/0.05 = - 0.8 For Mary, X =0.76, average = 0.83 and standard deviation = 0.08. Z score = (X - mean)/s.d. = (0.76 - 0.83)/0.08 = - 0.875

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    Jason’s score is 0.8 standard deviations below the mean. Mary’s score is 0.875 standard deviation below the mean. 8. A gumball machine contains 10 blue, 5 pink, 5 orange, 16 yellow and 14 white gumballs. What is the probability that a child receives a yellow followe d by a white? (No replacement). SHOW YOUR WORK Solution: Blue = 10 Pink = 5 Orange = 5 Yellow = 16 White = 14 Total = 50 Thus, P(Yellow) = (16/50) P(White) = (14/49) Required probability = P(Yellow )P(White) = (16/50)*(14/49) = 0.0914 9. Find the probability of rolling two dice and getting a sum of eight. SHOW/EXPLAIN YOUR WORK Solution: Sample space on rolling 2 dice is: { (1,1), (1,2,), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,3), (3,2,), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,5), (5,2,), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) } Total number of events in sample space = 36 Possible outcomes for getting sum of eight on rolling 2 dice is: {(2,6), (3,5), (4,4), (5,3), (6,2)} Thus, favorable number of outcomes = 5

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