Today’s Agenda

  • Forming Hypotheses
  • How rare is rare enough

Forming Hypotheses

Example:

An employee is excited to get a job at a company that has a fairly even distribution of men and women working at the company. However, the employee starts to notice that more men are obtaining promotions than women and thinks there is gender bias associated to whether someone gets a promotion or not. They want to test if there is statistically significant evidence that more men are getting promotions than women. Let \(p\) be the proportion of of people getting promotions that are men:

$H_0: $

$H_A: $

Possible Hypotheses
\(H_0\) \(H_A\)
\(=\) \(\neq\), \(>\), or \(<\)
\(\geq\) \(<\)
\(\leq\) \(>\)

How rare is rare enough?

Example:

Promotion Data

Promoted Not Promoted Total
Male \(21\) \(3\) \(24\)
Female \(14\) \(10\) \(24\)
Total \(35\) \(13\) \(48\)
  1. Assuming the null hypothesis is true, what is the \(z\)-score of the observed data?

  1. What percentage of data is as rare or more rare than the observed data?

  2. What conclusion can be made?

  1. Suppose a baker claims that his cookie diameter is more than \(10\) cm, on average. Several of his customers do not believe him. To persuade his customers that he is right, the baker decides to do a hypothesis test. He bakes \(10\) cookies. The mean diameter of the sample is \(12\) cm. The baker knows from baking hundreds of cookies that the standard deviation for the diameter is \(0.5\) cm. and the distribution of diameters is normal. Perform a hypothesis test with a \(5 \%\) significance level.

\(H_0=\)

\(H_A=\)

  1. Some people claim that they can tell the difference between a diet soda and a regular soda in the first sip. A researcher wanting to test this claim randomly sampled \(80\) such people. He then filled \(80\) plain white cups with soda, half diet and half regular through random assignment, and asked each person to take one sip from their cup and identify the soda as diet or regular. \(53\) participants correctly identified the soda. Does this data provide strong evidence that people can tell the difference between regular and diet soda? (when a significance level is not specified use \(\alpha=.05\)).

\(H_0=\)

\(H_A=\)

  1. It is believed that \(40 \%\) of people pass their driving test on the first attempt. Suppose you think the percentage is greater than \(40 \%\). So, you perform a hypothesis test and sample \(100\) people. Of the sampled people, \(47\) reply that they passed on their first attempt. Set up a hypothesis test and make a conclusion with a \(10 \%\) significance level.
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