Consider This… I claim that I make 80% of my basketball free throws. To test my claim, you ask me to shoot 20 free throws. I make only 8 out of 20. You decide to reject my claim. Why is a confidence interval not appropriate in this setting? Why do you reject my claim? Bad luck or a real difference? The Basic Idea: an outcome that would rarely happen if a claim were true is good evidence that the claim is not true.

Chapter 9: Testing a Claim 9.1 – Significance Tests: The Basics

How many licks does it take to get to the tootsie roll center of a tootsie roll pop?

So how long does an Everlasting Gobstopper really last? According to ChaCha, a Gobstopper should last 2 – 3 minutes. Let’s meet in the middle and say a gobstopper should last 2.5 minutes, on average. Do you believe this claim is true?

Stating Hypotheses The claim tested by a statistical test is called the null hypothesis (H0). The test is designed to assess the strength of the evidence against the null hypothesis. Often the null hypothesis is a statement of “no difference.” For our gobstopper example: 𝐻𝑜:  = 2.5

Stating Hypotheses The claim about the population that we are trying to find evidence for is the alternative hypothesis (Ha). For our gobstopper example: 𝐻𝑎:  > 2.5 𝐻𝑎:  < 2.5 𝐻𝑎:  ≠ 2.5 One-sided Two-sided

Stating Hypotheses: TA Scores The examinations in a large accounting class are scaled after grading so that the mean score is 50. The professor thinks that one TA is a poor teacher and suspects that his students have a lower mean score than the class as a whole. The TA’s students this semester can be considered a sample from the population of all students in the course, so the professor compares their mean with 50.

Stating Hypotheses: Spindle Diameter The diameter of a spindle in a small motor is supposed to be 5 mm. If the spindle is either too small or too large, the motor will not work properly. The manufacturer measures the diameter in a sample of motors to determine whether the mean diameter has moved away from the target.

Stating Hypotheses: Free Throws I claim that I make 80% of my basketball free throws. To test my claim, you ask me to shoot 20 free throws. I make only 8 out of 20. Am I overstating my abilities?

The air pressure of the basketballs produced by Wilson is supposed to be 15 pounds. If the balls are either over inflated or under inflated they will not bounce at the same level. Wilson measures the air pressure in a sample of balls to determine whether the mean pressure has moved away from the target.

During the 2009 major league baseball season, the average salary was $3,328,894. Members of the media suspect that the mean has increased for the 2010 season.

Last year, Vista football players spent an average of 122 minutes a week studying film. Do this year’s data show a different average length of time studying film?

A researcher claims that the proportion of college athletes who hold part-time jobs now is higher than the proportion known to hold such jobs a decade ago.

Coach Fox is concerned about his “negatives” – the percentage of Broncos fans who express disapproval of his job performance. His coaching staff decides to pays for a series of TV ads, hoping that they can keep the negatives below 20%. They will use follow-up polling to assess the ads’ effectiveness.

Is a coin fair?

So how long does an Everlasting Gobstopper really last? Let’s collect our data. Use the timer on your phone (or the clock) to see how long your Gobstopper lasts. No chewing! Record your time on the dotplot (to the nearest quarter of a minute)

So how long does an Everlasting Gobstopper really last? Now consider our hypotheses. Discuss with your neighbor – is ChaCha’s claim of 2.5 minutes reasonable? Why or Why not?

Interpreting P-Values The null hypothesis H0 states the claim that we are seeking evidence against. The probability that measures the strength of the evidence against a null hypothesis is called a p-value. Definition: The probability, computed assuming H0 is true, that the statistic would take a value as extreme as or more extreme than the one actually observed is called the P-value of the test. The smaller the P-value, the stronger the evidence against H0 provided by the data. Small P-values are evidence against H0 because they say that the observed result is unlikely to occur when H0 is true. Large P-values fail to give convincing evidence against H0 because they say that the observed result is likely to occur by chance when H0 is true.

Interpreting P-Values: Free Throws 𝑝 = the true proportion of free throws I make 𝐻 𝑜 :𝑝=0.8 𝐻 𝑎 :𝑝<0.8 An outcome that would occur so rarely by random chance when 𝐻 𝑜 is evidence against 𝐻 𝑜 . Therefore, we reject 𝐻 𝑜 :𝑝=0.8 and conclude the true proportion of free throws I make is less than 80%. I claim that I make 80% of my basketball free throws. To test my claim, you ask me to shoot 20 free throws. I make only 8 out of 20. Am I overstating my abilities? This gives a P-value of 0.000004.

Interpreting P-Values: TA Scores The examinations in a large accounting class are scaled after grading so that the mean score is 50. The professor thinks that one TA is a poor teacher and suspects that his students have a lower mean score than the class as a whole. In a sample of 30 of this TA’s students, the mean score is 49.2 with a standard deviation of 8. This gives a P- value of 0.3177. 𝜇 = the true mean score 𝐻 𝑜 :𝜇=50 𝐻 𝑎 :𝜇<50 An outcome that would occur so often by random chance (almost 1 in 3) when 𝐻 𝑜 is evidence for 𝐻 𝑜 . Therefore, we fail to reject 𝐻 𝑜 :𝜇=50 and conclude the true mean score for this TA is possibly 50. What P-value is the “cut-off” point for enough evidence?

Statistical Significance There is no rule for how small a P-value we should require in order to reject H0 — it’s a matter of judgment and depends on the specific circumstances. But we can compare the P-value with a fixed value that we regard as decisive, called the significance level. We write it as α, the Greek letter alpha. When our P-value is less than the chosen α, we say that the result is statistically significant. Definition: If the P-value is smaller than alpha, we say that the data are statistically significant at level α. In that case, we reject the null hypothesis H0 and conclude that there is convincing evidence in favor of the alternative hypothesis Ha. When we use a fixed level of significance to draw a conclusion in a significance test, P-value < α → reject H0 → conclude Ha (in context) P-value ≥ α → fail to reject H0 → cannot conclude Ha (in context)

Example: Better Batteries A company has developed a new deluxe AAA battery that is supposed to last longer than its regular AAA battery. Based on years of experience, the company knows that its regular AAA batteries last for 30 hours of continuous use, on average. H0 : µ = 30 hours Ha : µ > 30 hours where µ is the true mean lifetime of the new deluxe AAA batteries. The resulting P-value is 0.0276. What conclusion can you make for the significance level α = 0.05? Since the P-value, 0.0276, is less than α = 0.05, the sample result is statistically significant at the 5% level. We have sufficient evidence to reject H0 and conclude that the company’s deluxe AAA batteries last longer than 30 hours, on average. b) What conclusion can you make for the significance level α = 0.01? Since the P-value, 0.0276, is greater than α = 0.01, the sample result is not statistically significant at the 1% level. We do not have enough evidence to reject H0 in this case. therefore, we cannot conclude that the deluxe AAA batteries last longer than 30 hours, on average.

So how long does an Everlasting Gobstopper really last? Interpret the P-value for our data on how long a Gobstopper lasts. Are our results statistically significant?

Homework