1.

(a) For each of the following two cases, draw a diagram to show the relative positions of the main measures of central tendency: (i ) a symmetric distribution; (ii ) a right-skewed distribution. (b) Suppose that out of all UCD students, 40% have a UCD scarf, 25% have a UCD hoodie and 10% have both. Let A be the event that a randomly- chosen UCD student has a UCD scarf and let B be the event that he/she has a UCD hoodie. (i ) Are the events A and B mutually exclusive? Justify your answer in at most one sentence. (ii ) Are the events A and B independent? Justify your answer in at most one sentence. (c) You have five cards with the numbers 1, 2, 3, 4, 5, respectively, printed on them. You put the five cards into a hat. You draw a card, record the number on it, and replace the card. You then draw again and also record the second number you see. Let X be the total of the two numbers you recorded. Give the probability distribution for X as a table, showing each possible value for X together with the probability of X taking that value.

2. (a) Give two reasons why Data Analysis is relevant in business. Use at most one sentence for each reason. (b) Historical data shows that in the small claims court, before all judges except [ one, litigants have an equal chance of winning or losing; the exception is in cases heard by Judge Dredd, where the chance of winning is 75%. On average, Judge Dredd hears 10% of all cases. A friend of yours has just won a case at the small claims court but she does not remember the name of the judge. What is the probability that this case was heard by Judge Dredd?

3.

Your friend Joey has just been made manager of the boutique coffeeshop ArabicaRobusto. He has ideas for how to improve the business but wants to base any changes he makes on evidence. Given your knowledge of data analysis and probability, he asks for your advice. (a) Joey has asked customers to show on their way out how much they liked their visit to ArabicaRobusto, by tapping one of five buttons labelled 1, 2, 3, 4, 5, where a higher number tapped means a higher level of satisfaction. Today's data are: 1, 4, 2, 3, 3, 1, 2, 5, 3. (i ) Calculate the mean of this data set. (ii ) Calculate the standard deviation of this data set. (iii ) Decide whether or not this data set is skewed and interpret its meaning for Joey. (b) Joey has been studying numbers of customers arriving, to determine whether he should employ extra counter staff. He finds that, on average, 104 people arrive every hour. Joey has decided that he will open a new service point if the probability of at least two people arriving per minute is 50% or more. What do you recommend? Show all the steps of your work and explain your reasoning. (c) The shop sells three sizes of coffee: Grande, Grande+ and Grande++. Joey shows you their historical sales data: 50% of all coffees sold are Grande, 30% are Grande+ and 20% are Grande++. A group of ten people comes into the coffee shop. What is the probability that at least two of these ten people order a Grande++? Show all the steps of your work and explain your reasoning. (d ) You overhear one of the group of ten people in (c) say that she is definitely going to order the Grande++. Given this extra information, what is the probability that at least two of the ten people order a Grande++? Show all the steps of your work and explain your reasoning.

4.

Based on a research study looking at how people spend their time, it seems that the most recurrent thoughts are about food. Assuming the process is normally distributed, scientists have now selected a sample of 61 peo- ple whose average time spent thinking about food is 41.4 minutes per day with standard deviation of 4.2 minutes per day. Historical data reporting an average of 40 minutes per day spent thinking about food. Perform a hypothesis test for the population mean with a level of significance of 0.01 aimed at evaluating if there has been an increase in food-related thoughts. Answer the following: (a) What is the null hypothesis? (b) What is the alternative hypothesis? (c) What is the type of test (i. e., left-tailed/right-tailed/two-tailed)? (d ) What is the distribution that you will consider to compute the test statis- tic? Please provide motivation(s) for your choice. (e) What is the value of the test statistic? (f ) Please perform the critical value approach: (i ) What is/are the critical value(s)? (ii ) What does/do the critical value(s) represent in this case? (iii ) What is the rejection rule to be verified in this case? (iv ) Plot the rejection rule. (g) Please perform the p-value approach: (i ) What is the p-value? (ii ) What does the p-value represent in this case?

(iii ) What is the rejection rule to be verified in this case? (iv ) Plot the rejection rule. (h) What is the outcome of the test (i. e., do you reject or not reject the null hypothesis)? Please provide an interpretation of the final result.

5. (a) Explain in your own words what type of information a hypothesis test for one population mean provides. (b) List the assumptions underpinning a one mean z-test. (c) What is the difference in the assumptions between a one mean z-test and a one mean t-test? (d ) Explain in your own words the difference between non-parametric and para- metric hypothesis tests and provide an example for each category 6. (a) What is the generic structure of the null and alternative hypotheses for the goodness-of-fit test? (b) If you have 7 categories, what is the number of degrees of freedom? (c) If the value of the chi-square statistic is 21, assuming a level of significance of 0.05, what is the critical value or the p-value to consider? (d ) Based on point (c), please provide the test decision and plot the rejection rule.