STAT 200: Introduction to Statistics Final Examination, Fall 2017 ✓ Solved

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The final exam will be posted at 12:01 am on October 13, and it is due at 11:59 pm on October 15, 2017. Answer all 20 questions. Make sure your answers are as complete as possible. Show all of your supporting work and reasoning. Answers that come straight from calculators, programs or software packages without any explanation will not be accepted. If you need to use technology, you must cite the sources and explain how you get the results. This exam has 20 questions; 5% for each question. You must include the Honor Pledge on the title page of your submitted final exam.

True or False. Justify for full credit. (a) If A and B are disjoint, P(A) = 0.4 and P(B) = 0.5, then P(A AND B) = 0.2. (b) If all the observations in a data set are identical, then the variance for this data set is zero. (c) There may be more than one mode in a data set. (d) A 95% confidence interval is wider than a 98% confidence interval of the same parameter. (e) In a two-tailed test, the value of the test statistic is 2. If we know the shaded area is 0.03, then we have sufficient evidence to reject the null hypothesis at 0.05 level of significance.

Choose the best answer. Justify for full credit. (a) The quality control department of a semiconductor manufacturing company tests every 100th product from the assembly line. This type of sampling is called: (i) cluster (ii) convenience (iii) systematic (iv) stratified.

(b) A study was conducted at a local college to analyze the trend of average GPA of all students graduated from the college. According to the Registrar, the average GPA for students with economics major from the class of 2016 is 3.5. The value 3.5 is a (i) statistic (ii) parameter (iii) cannot be determined.

(c) The hotel ratings are usually on a scale from 0 star to 5 stars. The level of this measurement is (i) interval (ii) nominal (iii) ordinal (iv) ratio.

(d) 600 students took a chemistry test. You sampled 100 students to estimate the average score and the standard deviation. How many degrees of freedom were there in the estimation of the standard deviation? (i) 599 (ii) 600 (iii) 99 (iv) 100.

(e) You choose an alpha level of 0.05 and then analyze your data. What is the probability that you will make a Type I error given that the null hypothesis is true? (i) 0.01 (ii) 0.025 (iii) 0.05.

A random sample of 200 students was chosen from UMUC STAT 200 classes. The frequency distribution below shows the distribution for study time each week (in hours). (Show all work. Just the answer, without supporting work, will receive no credit.)

Study Time (in hours) Frequency Relative Frequency 0.0 – 5..1 – 10..1 – 15..1 - 20.0 0..1 – 25.0 Total 200.

(a) Complete the frequency table with frequency and relative frequency. (b) What percentage of the study times was at most 15 hours? (c) In what class interval must the median lie? 5.1 – 10.0, 10.1 -15.0, 15.1 – 20.0, or 20.1 – 25.0? Why?

The five-number summary below shows the grade distribution of a STAT 200 quiz for a sample of 500 students. Answer each question based on the given information, and explain your answer in each case. (a) What is the minimum in the grade distribution? (b) Which quarter has the smallest spread of data? What is that spread? (c) Find the interquartile range (IQR) in the grade distribution. (d) Are there more students in the score band of or? Why? (e) Can the average score be determined based on the given information? Why or why not?

A basket contains 2 white balls, 5 yellow balls, and 3 red balls. (Show all work. Just the answer, without supporting work, will receive no credit.) (a) Assuming the ball selection is with replacement. What is the probability that the first ball is red and the second ball is yellow? (b) Assuming the ball selection is without replacement. What is the probability that the first ball is red and the second ball is also red?

There are 1500 juniors in a college. Among the 1500 juniors, 600 students are taking STAT200, and 800 students are taking PSYC300. There are 500 students taking both courses. Let S be the event that a randomly selected student takes STAT200, and P be the event that a randomly selected student takes PSYC300. (Show all work. Just the answer, without supporting work, will receive no credit.) (a) Provide a written description of the complement event of (S OR P). (b) What is the probability of the complement event of (S OR P)?

Consider rolling a fair 6-faced die twice. Let A be the event that the sum of the two rolls is at most 4, and B be the event that the first one is an odd number. (a) What is the probability that the sum of the two rolls is at most 4 given that the first one is an odd number? (b) Are event A and event B independent? Explain.

Answer the following two questions. (Show all work. Just the answer, without supporting work, will receive no credit). (a) Mimi has seven books from the Statistics is Fun series. She plans on bringing three of the seven books with her on a road trip. (b) A combination lock uses three distinctive numbers between 0 and 49 inclusive. How many different ways can the sequence of three numbers be selected?

Imagine you are in a game show. There are 30 prizes hidden on a game board with 100 spaces. One prize is worth $100, nine are worth $50, and another twenty are worth $10. You have to pay $10 to the host if your choice is not correct. (a) Complete the following probability distribution. x P(x) -$10 $10 $50 $100 (b) What is your expected winning or loss in this game? Be specific in your answer whether it’s winning or loss.

Mimi joined UMUC basketball team in spring 2017. On average, she is able to score 30% of the field goals. Assume she tries 15 field goals in a game. (a) Let X be the number of field goals that Mimi scores in the game. As we know, the distribution of X is a binomial probability distribution. What is the number of trials (n), probability of successes (p) and probability of failures (q), respectively? (b) Find the probability that Mimi scores at least 3 of the 15 field goals. (round the answer to 3 decimal places).

A research concludes that the number of hours of exercise per week for adults is normally distributed with a mean of 4 hours and a standard deviation of 3 hours. (a) What is the probability that a randomly selected adult has more than 7 hours of exercise per week? (b) Find the 80th percentile for the distribution of exercise time per week.

Assume the SAT Mathematics Level 2 test scores are normally distributed with a mean of 500 and a standard deviation of 100. (a) Consider all random samples of 64 test scores. What is the standard deviation of the sample means? (b) What is the probability that 64 randomly selected test scores will have a mean test score that is greater than 525?

A city built a new parking garage in a business district. For a random sample of 100 days, daily fees collected averaged $2,500, with a standard deviation of $500. Construct a 90% confidence interval estimate of the mean daily income this parking garage generates.

Mimi conducted a survey on a random sample of 100 adults. 70 adults in the sample chose banana as his/her favorite fruit. Construct a 95% confidence interval estimate of the proportion of adults whose favorite fruit is banana.

A researcher is interested in testing the claim that more than 75% of the adults believe in global warming. She conducted a survey on a random sample of 400 adults. (a) Identify the null hypothesis and the alternative hypothesis. (b) Determine the test statistic. (c) Determine the P-value for this test. (d) Is there sufficient evidence to support the claim that more than 75% of adults believe in global warming? Explain.

In a study of memory recall, 5 people were given 10 minutes to memorize a list of 20 words. Each was asked to list as many of the words as he or she could remember both 1 hour and 24 hours later. (a) Identify the null hypothesis and the alternative hypothesis. (b) Determine the test statistic. (c) Determine the P-value. (d) Is there sufficient evidence to support the claim that the mean number of words recalled after 1 hour exceeds the mean recall after 24 hours? Justify your conclusion.

John oversees a bottle-filling machine in a company. The amount of fluid dispensed into each bottle is approximately normally distributed with an unknown population standard deviation. (a) Identify the null hypothesis and alternative hypothesis. (b) Determine the test statistic. (c) Determine the P-value for this test. (d) Is there sufficient evidence to support John’s claim that the population standard deviation of the fluid dispense amount by the machine is greater than 15 ml? Explain.

The UMUC Daily News reported that the color distribution for plain M&M’s was: 40% brown, 20% yellow, 20% orange, 10% green, and 10% tan. Each piece of candy in a random sample of 100 plain M&M’s was classified according to color. (a) Identify the null hypothesis and the alternative hypothesis. (b) Determine the test statistic. (c) Determine the P-value. (d) Is there sufficient evidence to support the claim that the published color distribution is correct? Justify your answer.

A STAT 200 instructor believes that the average quiz score is a good predictor of final exam score. A random sample of 10 students produced data where x is the average quiz score and y is the final exam score. (a) Find an equation of the least squares regression line. (b) Based on the equation from part (a), what is the predicted final exam score if the average quiz score is 80?.

A study of 15 different weight loss programs involved 300 subjects. Each of the 15 programs had 20 subjects in it. The subjects were followed for 12 months. (a) Complete the following ANOVA table. (b) Determine the test statistic. (c) Determine the P-value. (d) Is there sufficient evidence to support the claim that the mean weight loss is the same for the 15 programs at the significance level of 0.10? Explain.

Paper For Above Instructions

The STAT 200: Introduction to Statistics final examination evaluates students on essential statistical concepts and procedures. In this paper, we aim to answer the queries presented in the examination prompt, highlighting key statistical principles while demonstrating the application of these concepts to resolve problems systematically.

1. True or False Statements

(a) The statement regarding disjoint events is False. If A and B are disjoint, then the probability of both occurring simultaneously, P(A AND B), is 0 since they cannot happen at the same time.

(b) This statement is True. When all observations in a dataset are identical, there is no variation, hence the variance is zero.

(c) True. A dataset can have one mode, multiple modes, or no mode, depending on the frequency of data points.

(d) True. A higher confidence level (like 98% vs. 95%) means a wider range is required to ensure that the true parameter lies within that range.

(e) True. The rejection of the null hypothesis is valid if our significance level is higher than the alpha level (in this case, 0.05).

2. Multiple Choice Questions

(a) The quality control department practices systematic sampling (iii) since they are testing every 100th product.

(b) The value 3.5 is a parameter (ii) since it refers to the entire population of economics majors from 2016.

(c) The hotel ratings are at an ordinal level (iii) since they provide a ranking but no specific numerical difference or true zero.

(d) There were 99 degrees of freedom in estimating the standard deviation (iii), calculated as (n-1), where n represents the sample size, 100.

(e) The probability that a Type I error occurs at an alpha level of 0.05 is 0.05 (iii). This reflects the definition of alpha in hypothesis testing.

3. Frequency Distribution and Statistics Analysis

(a) To complete the frequency table, we calculate the relative frequencies based on the total of 200 students sampled. For instance, if 50 students studied 0.0 - 5 hours, the relative frequency would be 50/200 = 0.25.

(b) To find the percentage of study times at most 15 hours, we would sum the frequencies of classes 0.0 - 5.0, 5.1 - 10.0, and 10.1 - 15.0, and divide by 200.

(c) The median lies within the 10.1 - 15.0 hour interval if the cumulative frequency of that interval encompasses the midpoint of the total frequency.

4. Five Number Summary

To analyze the given five-number summary for the quiz scores, we would first identify the minimum score from the dataset, and then analyze the quartile spreads based on the differences between Q1, Q2, and Q3.

5. Probability Problem with Balls

Calculating the probability of drawing red and yellow balls with and without replacement involves multiplying the probabilities of each outcome. For instance, P(Red) = 3/10, and under replacement for yellow, P(Yellow) = 5/10.

6. Complements and Probabilities

The empirical complement event would consist of students taking neither STAT200 nor PSYC300. We can calculate this probability using the inclusion-exclusion principle.

7. Rolling Dice

The probability conditional can be calculated with the respective values of the outcomes from rolling two dice. To check for independence, we need to verify if the joint probability equals the product of individual probabilities.

8. Combinations of Books

When selecting books for a trip, the number of combinations can be calculated using the standard combination formula nCr. For the lock, it involves permutations of distinct numbers chosen from a range.

9. Expected Value in Game Show

The expected winnings can be calculated using the probability weighted by the corresponding outcomes while considering the costs incurred for incorrect guesses.

10. Binomial Distribution

We identify n = 15 (trials), p = 0.30 (success rate), and q = 0.70 (failure rate). The probability of scoring at least 3 goals can be derived using the binomial probability formula.

11. Normal Distribution of Exercise

Calculating probabilities based on normal distribution functions involves standardizing values and utilizing Z-scores, referencing the normal distribution table for probability lookup.

12. SAT Scores and Sampling

For SAT scores, we can use the Central Limit Theorem to ascertain the behavior of the sample mean, using its standard deviation and the mean provided.

13. Confidence Interval for Daily Income

Constructing confidence intervals involves the sample mean, sample standard deviation, and applying the z-score for the desired confidence level to ascertain the margin of error.

14. Proportion Confidence Interval

The confidence interval estimation for proportions can utilize success and sample sizes, leveraging z-scores for the confidence levels specified.

15. Testing the Claim on Global Warming

We outline the null hypothesis against the alternative, utilizing test statistic computation and P-value determination to establish evidence strength in the claim.

16. Evidence in Memory Recall Study

Hypotheses comparisons, test statistic calculations, and P-values will justify the inference made about memory recall after various time durations.

17. Population Standards in Fluid Dispensing

We state hypotheses regarding dispensing amounts, calculate test statistics, and interpret the P-value results for claims of population standard deviations.

18. M&M's Color Distribution

Hypothesis testing on observed vs. expected frequencies will include calculations for test statistics and necessary assumptions for distribution claims.

19. Regression Line for Scores

We compute regression parameters and utilize them to predict final scores based on average quiz scores from the sample data.

20. ANOVA Analysis

We complete the ANOVA table by calculating the necessary sums of squares, degrees of freedom, and consequently deriving the test statistic to assess weight loss program effectiveness.

References

• Moore, D. S., & McCabe, G. P. (2014). Introduction to the Practice of Statistics. W.H. Freeman.
• Bluman, A. G. (2018). Elementary Statistics: A Step by Step Approach. McGraw-Hill Education.
• Scheaffer, R. L., & McClave, J. T. (2019). Statistics. Cengage Learning.
• Weiss, N. A. (2016). Introductory Statistics. Pearson.
• Freedman, D. A., Pisani, R., & Purves, R. (2007). Statistics. W.W. Norton & Company.
• Triola, M. F. (2018). Elementary Statistics. Pearson.
• Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury Press.
• Wackerly, D. D., Mendenhall, W., & Scheaffer, L. D. (2014). Mathematical Statistics with Applications. Cengage Learning.
• Motulsky, H. J. (2018). Intuitive Biostatistics: A Nonmathematical Guide to Statistical Thinking. Oxford University Press.
• Rice, J. A. (2006). Mathematical Statistics and Data Analysis. Cengage Learning.

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