Topic

The use/misuse of P-values

Lecture

topic 2, The 6 steps of hypothesis testing

topic 6, Some practical hypothesis testing remarks

Purpose

Practice working with P-values. Let's do it!

Scenario

Suppose you just finished your studies at HEC Paris and you landed this fantastic job. It pays well. On the top of that you will receive a lucrative quaterly bonus if your work improves customer satisfaction. Your manager receives once a quarter the results from a random sample of customers who indicate their satisfaction with the company's product and services on a 0-100 scale.

Your first evaluation moment (and bonus?) is almost there! The target is to improve over the average satisfaction from last year, which was 75. There comes the email from your manager. You open it, and find that the average satisfaction is... 79!

Question 1

Of course, you deserve your first bonus! But you also know that your manager is strong with statistics. In your own words, is your first bonus a done deal based on the information provided in the scenario above? Why? Why not?

Question 2

Your manager computed the P-value and lets you know that the P-value is 0.0004. Can you now celebrate? Why? Why not?

Question 3

Suppose that the P-value was 0.88 instead. Does your conclusion change? Why? Why not?

Question 4

Suppose the P-value was 0.049. Should you get your bonus? Why? Why not?

Question 5

Suppose that the sample produced an average satisfaction of 75.8 and a P-value of 0.002. Would you celebrate? Why? Why not?

Topic

Hypothesis test for a mean

Lecture

topic 2, The 6 steps of hypothesis testing -- for a population mean --

Purpose

Practice carrying out the 6 steps of hypothesis testing for a mean by hand. Let's do it!

Topic

Hypothesis test for (a) proportion(s)

Lecture

topic 3, Hypothesis test for a proportion

topic 4, Hypothesis test for proportionS

Purpose

Practice carrying out the 6 steps of hypothesis testing for one or more proportions by hand. Let's do it!

Question 1

Use a $\chi^{2}$-test for the scenario above.

Question 2

Use a t-test for the scenario above to test an hypothesis about $\pi$, where $\pi$ indicates the proportion of students that prefers the 9am class.

Topic

Sample size calculations

Lecture

topic 5, Sample size calculations

Purpose

Practice calculating the sample size needed for making a reliable decision or conclusion. Let's do it!

Topic

Which hypothesis and which test statistic?

Lecture

topic 2, The 6 steps of hypothesis testing -- for a population mean --

topic 3, Hypothesis test for a proportion

topic 4, Hypothesis test for proportionS

Purpose

Practice identifying the correct null and alternative hypotheses and test statistic. Let's do it!

Question

Suppose you obtained a random sample from the national bureau of statistics with information about residents (adults) in France (to keep it simple, assume that one row in the data file is a French resident). The data covers 2020. You observe for instance wether someone is married, the persons income, the person's gender etc. (these are columns in your data file). Your manager wants to compare the situation in 2020 to the situation in 2010. Which hypothesis tests would you recommend for each one of the following cases? Write down the null and alternative hypothesis (step 1) and the formulate for the test statistic (step 3).

Case 1

Did the proportion of married people change from 2010? Assume married is capture as 1=married and 2=not married, and that the proportion of married people in 2010 was 0.53.

Well, what did you write down? Here we have a categorical variable, so we are talking about proportions. In the population, proportion is represented by the symbol $\pi$, and hypotheses always need to be stated in population terms. Here we are wondering whether the proportion of married people changed from 0.53. So, we would write $H_{0}: \pi=0.53$ and $H_{1}: \pi \not= 0.53$. We can use the Z-test statistic for this question, but we need to take the correct one (i.e. the one for categorical data), which is $Z= \frac{p-\pi}{\sqrt{\pi(1-\pi)/n}}$. Do you remember what $p$ stands for? Note that we could potentially also use the $\chi^{2}$-statistic here, in that case I would write the null hypothesis as $H_{0}: \pi_{1}=0.53, \pi_{2}=0.47$, where 1=married, 2=not married. See also item 3 of this app.

Case 2

Did income grow, where the average income in 2010 was 34000 euros?

Here, we are dealing with a quantitative variable, and we learned about testing an hypothesis about the population mean $\mu$ for such a scenario. The null hypothesis has to be a statement about the population, i.e. a Greek letter, and here we wonder whether the mean is bigger than 34000 euros. So, $H_{0}: \mu=34000$ and $H_{1}: \mu>34000$. Here it would also be fine to write down $H_{1}: \mu \not= 34000$, which would give a more conservative test (meaning, harder to reject the null hypothesis so stronger evidence from the data is needed to reject $H_{0}$). As for the previous question, we would use a Z-test statistic, but now for a quantitative variable, i.e. $Z= \frac{\bar{x}-\mu}{S / \sqrt{n}}$, where $\bar{x}$ is the sample mean and $S$ is the sample standard deviation both computed from the 2020 data.

Case 3

Whether employment status changed, where employment status is measured as 1=employed (excl. self-employed), 2=self-employed, 3 = unemployed, 4=other (e.g. retired). In 2010, the percentages were 62, 10, 10, and 18, respectively.

Employment status is a categorical variable with four categories. Like case 1, we again need to test about (population) proportions. We have four $\pi$'s in the null hypothesis. More specifically, $H_{0}: \pi_{1} = 0.62, \pi_{2} = 0.10, \pi_{3} = 0.10, \pi_{4} = 0.18$, the subscripts are defined above in the question text. Note that the $\pi$'s are proportions not percentages! For the alternative hypothesis we could write $H_{1}:$ at least one $=$ is $\not=$. Here, we have to use the $\chi^{2}$-test statistic, which is given by $\chi^{2} = (O_{1}-E_{1})^{2}/E_{1} + (O_{2}-E_{2})^{2}/E_{2} + (O_{3}-E_{3})^{2}/E_{3} + (O_{4}-E_{4})^{2}/E_{4}$; the $O$'s are the observed frequencies in the 2020 sample (e.g. $O_{1}$ is the number of people in your sample being employed) and the $E$'s are computed as the proportions under $H_{0}$ times the sample size.

Case 4

Are people travelling more or less often internationally compared to 2010? Assume that international travel is measured in your dataset as the exact number (0,1,2,3,. etc. times), and that the average number of international trips in 2010 was 3.8.

As always, first identify the measurement level of the variable in question. Here, international travel is measured as the exact number, so it is a quantitative variable. That means we can compute the mean and the hypothesis will be about the population mean $\mu$. More specifically, given the situation in 2010, the null hypothesis is $H_{0}: \mu=3.8$ and the alternative hypothesis is $H_{1}: \mu \not= 3.8$. The test statistic is the same as in scenario 2.

Topic

Practice working with SPSS

Lecture

topic 2, The 6 steps of hypothesis testing -- for a population mean --

topic 4, Hypothesis test for proportionS

This activity is only useful if you have reviewed the corresponding SPSS How to guide.

Purpose

Use SPSS to test hypotheses about the population. Follow the following managerial/research questions to further practice your SPSS skills. Let's do it!

In this module you learned about an hypothesis test for a mean and for proportions. SPSS can easily perform such tests for you (of course, you can also do them by hand, which is always a fun activity for a rainy Sunday afternoon!). It is always a good idea to write down the 6 steps on scratch paper, even when using SPSS. Practice conducting a t-test and a chi-square test with SPSS following two scenarios from the insurance claims and fraud mini-case (module 2, SPSS file 'mini_case_insurance_fraud_web.sav').

Question 1

Suppose the managers at the insurance company wanted to know whether the average claim amount this time period changed from previous time period. The accounting department reports that the average claim last year was $63500. What would your conclusion be?

Question 2

Similarly, management was concerned about the distribution of claim types this year. Knowing what claims to expect helps the insurance company to plan their risk. One manager argued that there was a trend that the Theft/Vandalism claims went down, as well as Contamination claims, whereas other claim types (in particular wind and hail damage) went up (relatively speaking). The manager prepared the data from last year: W/H 22%, Water 14%, F/S 23%, Contamination 10%, T/V 31%. Is there evidence in the data to support the manager's claim?

Topic

Five multiple choice practice questions

Lecture

Module 3, all topics

Purpose

Test your knowledge about the subjects of this module. Let's do it!