1. Chapter 8: Hypothesis Testing for Population Proportions
The basics of Significance Testing

2. Statistical Inference
Already discussed confidence intervals for unknown population parameter, p
CI’s used when the goal is to estimate an unknown population parameter like ρ
This chapter... statistical inference through significance tests
Evaluate evidence (a statistic) provided by sample data about some claim concerning an unknown population parameter like ρ

3. I’m a great free-throw shooter...
I claim that I make 95% of my basketball free throws.
To test my claim, I am asked to shoot 20 free throws. I make only 8 of the 20 (only 40%). Now people don’t believe that my claim of making 95% of my basketball free throws.
Making only 8 of 20 attempts would almost never happen/very unlikely if I truly did make 95% of my free throws
What would we expect if he/she truly did make 95% of all free throws? 19? 18? 20? 1?

4. Significance Testing
Basic idea... An outcome that would rarely happen if a claim were really true is good evidence that the claim is not true.
Example... I claim that 99% of adult humans are 6 feet tall or taller.
If my claim was true, it would be very rare to get most of the adult humans in a SRS of 100 that are shorter than 6 feet.

5. Significance Testing... Let’s begin by knowing μ and σ (unrealistic)
Because paramedic response time is critical to saving lives, several cities monitor these response times. In one city, the mean response time to all accidents involving life-threatening injuries last year was μ = 6.7 minutes with a standard deviation of σ = 2 minutes. The city manager encourages them to “do better” next year. At the end of the following year, the city manager selects a SRS of 400 calls involving life-threatening injuries. For this sample, the mean response time was 𝑥 = 6.48 minutes. Do these data provide good evidence that response times have decreased since last year?
This chapter is all on proportions; next on means. Some of my examples here will deal with proportions; some with means. These are just the basics that apply in either case.

6. Previous Year: μ = 6.7 minutes; σ = 2 minutes Following year, SRS 400 with 𝑥 = 6.48 minutes
Does this data provide good evidence that the response times have decreased? Remember, statistics vary from sample to sample. Maybe 𝑥 = 6.48 is a result of sampling variability. Maybe response time hasn’t improved.

7. Previous Year: μ = 6.7 minutes; σ = 2 minutes Following year, SRS 400 with 𝑥 = 6.48 minutes
Make a claim and see if the data provides evidence against it. Ho: μ = 6.7 minutes Ho, null hypothesis; usually no effect, no change, no difference; neutral Hypothesis always refers to some population parameter, like μ or ρ (NEVER a sample statistic! Don’t want to make hypothesis about something we already know.)
We will assume the null is true throughout our entire process, until the very last step. Kind of like presumed not guilty until proven otherwise.

8. Previous Year: μ = 6.7 minutes; σ = 2 minutes Following year, SRS 400 with 𝑥 = 6.48 minutes
Ho: μ = 6.7 minutes We are seeking evidence of a decrease in response time, so Ha: μ < 6.7 minutes Ha, alternative hypothesis, claim about population we are trying to find evidence for. One-sided, only interested in decrease (in this case) but can be two-sided, such as Ho: μ = 0 and Ha: μ ≠ 0

9. Is our sampling distribution’s 𝑥 rare?
Sampling dist is approx Normal (will get to this next chapter as to why). Our x bar is very rare. More than 3 SD’s away from assuming 6.7 was true, this 6.48 would very rarely happen; evidence that 6.7 is not true but rather it is less than 6.7 so Ha is true. Just trying to get the idea of what a Hyp test is.
If our x bar were say 6.61, that’s only less than 1 SD away from 6.7, so not so rare. Could easily happen. So not good evidence against Ho.

10. Practice: State the appropriate null hypothesis and alternative hypothesis in each case. Be sure to define your parameter each time.
Larry's car averages 26 miles per gallon on the highway. He switches to a new brand of motor oil that is advertised to increase gas mileage. After driving 3000 highway miles with the new oil, he wants to determine if the average gas mileage has increased.
Parameter: μ = mean gas mileage for Larry’s car on the highway
Ho: μ = 26 mpg Ha: μ > 26 mpg

11. Practice: State the appropriate null hypothesis and alternative hypothesis in each case. Be sure to define your parameter each time.
A May 2005 Gallup Poll report on a national survey of 1028 teenagers revealed that 72% of teens said they rarely or never argue with their friends. You wonder whether this national result would be true in your school. So you conduct your own survey of a random sample of students at your school.
Parameter: Proportion of teens in your school who rarely or never fight with their friends.
Ho: p = Ha: p ≠ 0.72

12. Explain what is wrong in each situation and why it is wrong
A change is made that should improve student satisfaction with the parking situation at your school. The null hypothesis, that there is an improvement, is tested versus the alternative, that there is no change.
Ho and Ha have been switched. The null hypothesis should be a statement of ‘no change.’

13. Explain what is wrong in each situation and why it is wrong
A researcher tests the following null hypothesis: H0: 𝑥 = 10.
The null hypothesis (and the alternative hypothesis as well) should be a statement/claim about a population parameter (like µ), not a sample statistic (like 𝑥 )
Or p and p hat.

14. Explain what is wrong in each situation and why it is wrong
The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures students' attitudes toward school and study habits. Scores range from 0 to 200. The mean score for U.S. college students is about 115. A teacher suspects that older students have better attitudes toward school.
Ho: μ = Ha: μ > 120
Ho and Ha must share same numeric value (only change =, >, <, ≠)

15. Explain what is wrong in each situation and why it is wrong
The Census Bureau reports that households spend an average of 31% of their total spending on housing. A homebuilders association in Cleveland believes that this average is lower in their area. They interview a sample of 40 households in the Cleveland metropolitan area to learn what percent of their spending goes toward housing. Take μ to be the mean percent of spending devoted to housing among all Cleveland households.
H0: p = 31% Ha: p < 31%
Nothing is wrong. Perfect.

16. Conditions for Significance Tests (just like Confidence Intervals)
SRS (randomization)
Normality (for means, proportions; requirements are different)
Independence (population must be --or must be able to reasonably assume that it is-- at least 10 times as large as the sample size; and that one observation has no influence on any others)
Means – population normal, sample is automatically normal. Pop not normal or unknown, then look for n >= 30 for CLT; if n < 30, need to do box plot or NPP to show fairly symmetric and no outliers
Proportions np >= 10 and n (1-p) >=10

17. Significance tests use test statistics...
Some principles that apply to most tests:
The test is based on a statistic that compares the value of the parameter (Ho : μ = ) with an estimate of the parameter from the sample data ( 𝑥 , 𝑝 )
Values of the estimate far from the parameter value in the direction specified by the alternative hypothesis give evidence against H0
To assess how far the estimate is from the parameter, standardize the estimate (what does this mean?)
Or could be Ho = p; talk about basketballs throws and paramedics examples. ‘standardize’ Minitab does this calculation automatically... But it is a z-score, # of SD away from (claimed) center.

18. Good evidence against μ = 6.7 minutes...
... But rather μ < 6.7 minutes this year. Data would unlikely happen if Ho were true. But how unlikely? Need precise way to measure ‘how unlikely.’

19. P-Values
A p-value is a quantitative measure of rarity of/how unlikely a finding Small p-values are evidence against Ho Large p-values fail to give evidence against Ho
P as probability P(x) =; p as population parameter; p as in p-value. Don’t get confused. P-value is basically area under the curve at that boundary or more severe.
Go on to next slide for definition; then the next slide is the graphic again.

20. Definition of a P-Value
The probability computed assuming that Ho is true, that the observed outcome would take a value as extreme as or more extreme than the actual observed value. Let’s go back to the paramedics example again...

21. Ho: μ = 6.7 Ha: μ < 6.7 z = -2.20
(Just fyi, in this example, negative values of z favor Ha over Ho (not always the case))
Remember, p-value is the probability.... Or the area under the curve... what’s the area under the curve for z = -2.20?

22. Ho: μ = 6.7 Ha: μ < 6.7 p-value = 0.0139
Small p-value  strong evidence against Ho
Favors alternative hypothesis, Ha: μ < 6.7 minutes

23. What’s the difference between...
Ho: p = 0.2 Ha: p < 0.2 and Ho: p = 0.2 Ha: p ≠ 0.2

24. If Ha is 2-sided (≠), both directions count
Because the alternative is two-sided, the P-value is the probability of getting a z at least as far from 0 in either direction as the observed z = 1.20.
Big p-value; good evidence against Ho

25. Statistical Significance...
Most of the time, we take one more step to assess evidence against Ho
We compare p-value to some pre-determined value (versus ‘unlikely’) called a significance level, symbol α (alpha)
Can think of this as a rejection zone (sketch)
Page 343 Your textbook explains it as representing the probability of rejecting the null hypothesis when the null is true. More on this later.

26. Statistical Significance
Significance level makes ‘not likely’ more exact, more informative
Most common α levels are α = 0.05 or α = 0.01
Interpretation:
At α = 0.05, data give evidence against Ho so strong it would happen no more than 5% of the time
Like 16 for CDL or 21 to drink and gamble

27. Statistical Significance
If p-value is as small or smaller than α, we say data are statistically significant at level α
Note: ‘significant’ in statistics doesn’t mean important (like in English); it means not likely to happen by chance

28. Statistically Significant Sketches
If p-value is p = this is significant at α = 0.05 level (in rejection zone)
If p-value is p = this is not significant at α = 0.01 level (not in rejection zone)
Draw pics.

29. Interpretation/Wording
Reject Ho (Null Hypothesis): This happens when sample statistic is statistically significant, p-value is too unlikely to have occurred by chance (we don’t believe null hypothesis), in the rejection zone Wording must reference all of the following for a complete interpretation... p-value, α level, reject Ho, and conclusion in context (caution about using the word ‘cause’ or ‘prove’).

30. Interpretation/Wording
Fail to Reject Ho (Null Hypothesis): This happens when sample statistic could have occurred by chance (we do believe null hypothesis; we don’t believe the alternative), not in rejection zone Wording must reference all of the following for a complete interpretation... p-value, α level, fail to reject Ho, and conclusion in context (caution about using the word ‘cause’ or ‘prove’)

31. Tests about a population proportion ...
Disregard means question; we will get to that next chapter.

32. Conditions for Tests about a population proportion...
Random Sample ... SRS or randomly selected or randomly assigned
Large Sample Size; Normality ... npo ≥ 10 and n(1 – po) ≥ 10
Independence ... Population at least 10 times sample size; and each observation has no influence on any other

33. Work stress...
According to the National Institute for Occupational Safety and Health, job stress poses a major threat to the health of workers. A national survey of restaurant employees found that 75% said that work stress had a negative impact on their personal lives. A simple random sample of 100 employees from a large restaurant chain finds that 68 answer “Yes” when asked, “Does work stress have a negative impact on your personal life?” Is this good reason to think that the proportion of all employees in this chain who would say “Yes” differs from the national proportion p0 = 0.75? H0: p = 0.75 Ha: p ≠ 0.75 We want to test a claim about p, the true proportion of this chain's employees who would say that work stress has a negative impact on their personal lives.

34. Work stress... Conditions: 1-proportion z test SRS – stated in problem
Normality - The expected number of “Yes” and “No” responses are (100)(0.75) = 75 and (100)(0.25) = 25, respectively. Both are at least 10.
Independence - Since we are sampling without replacement, this “large chain” must have at least (10)(100) = 1000 employees; and we must assume that one employee does not influence the response of any other employee

35. Work stress.... Calculations for 1-prop z test; use Minitab
1 sample, proportion; change options and data as needed
By Minitab, z = and p-val = ; sketch curve as visual

36. Work stress...
Interpretation: Fail to reject Ho. There is over a 10% (which is well over a reasonable α level) chance of obtaining a sample result as unusual as or even more unusual than we did ( = 0.68) when the null hypothesis is true. We have insufficient evidence to suggest that the proportion of this chain restaurant's employees who suffer from work stress is different from the national survey result, 0.75.
We will use reject or fail to reject (not accept); we fail to reject because we don’t have enough evidence to reject. Kind of like not guilty vs. innocent.

37. We want to be rich...
In a recent study, 73% of first-year college students responding to a national survey identified “being very well-off financially” as an important personal goal. A state university finds that 132 of an SRS of 200 of its first-year students say that this goal is important.
Is there evidence that the proportion of first-year students at this university who think being very well-off is important differs from the national value, 73%? Carry out a significance test to help answer this question.

38. n = 200; x = 132; SRS; p = .73; 𝑝 = 0.66
We want to test Ho: p = 0.73 versus Ha: p ≠ 0.73 regarding the proportion of first-year students at this university who think being very well-off is important differs from the national value of 73%

39. n = 200; x = 132; SRS; p = .73; 𝑝 = 0.66
Conditions: SRS – stated in problem Normality – np ≥ 10 & n (1 – p) ≥ 10 (200)(0.73) ≥ 10 & (200) ( ) ≥ 10 Independence – We must assume at least (10)(200) first-year students in the population and that one student’s response does not influence any other student’s response.

40. Interpretation...
Reject Ho. With a p-value of , and assuming an α = 0.05, we conclude that we do have statistically significant evidence that the proportion of all first-year students at this university who think being very well-off is important differs from the national value. (determination, p-value, α, and context... Always)
Z = -2.23

41. Use & Abuse of Tests...
Significance tests are used in a variety of settings... Marketing, FDA drug testing, discrimination court cases, etc.
Significance tests quantify event that is unlikely to occur simply by chance
Different levels of significance (α) are chosen depending on the given situation; typically α = 0.10, 0.05, or 0.01
We will use 5% alpha most of the time for this class.

42. Use & Abuse of Tests...
P-values allow us to decide individually if evidence is sufficiently strong
But, there is still no practical distinction between p-values of, say, and 0.051
Statistical inference does not correct basic flaws in survey or experimental design

43. Using Inference to Make Decisions...
Sometimes we do everything correctly... data collection, conditions, calculations, interpretation... but we still make an incorrect decision/determination... perhaps we just happen to get a sample statistic that is very extreme... that really doesn’t represent our population accurately ... we reject the null hypothesis when we really should have failed to reject (Ho was really true) OR we fail to reject the null hypothesis when we really should have rejected the null hypothesis (Ho was really false) ... we make an error

44. Making errors when using inference...
Type I Error
We reject Ho (null hypothesis) when Ho is really true
In other words, we determine Ha (alternative hypothesis) is true when, in actuality, Ho (null hypothesis) is true
Type II Error
We fail to reject Ho (null hypothesis) when Ho is really false
In other words, we determine Ho (null hypothesis) is true, when, in reality, Ha (alternative hypothesis) is true
Don’t need to know types; but many textbooks refer to these errors in this way.

45. Type I and Type II Errors...

46. Paramedic Response Times Revisited...
H0: μ = 6.7 minutes
Ha: μ < 6.7 minutes
... where μ was the mean response time to all calls involving life-threatening injuries this year.
Type I error: reject H0 when H0 is true
Description: The city manager concludes that the mean response time this year is less than 6.7 minutes (last year's average) when in fact the mean response time is still 6.7 minutes (or higher).
Consequences: The city manager believes that paramedic response times have improved when they really haven't. This could result in additional loss of life for accident victims.

47. Paramedic Response Times Revisited...
H0: μ = 6.7 minutes
Ha: μ < 6.7 minutes
... where μ was the mean response time to all calls involving life-threatening injuries this year.
Type II error: fail to reject H0 when H0 is false
Description: The city manager decides that the paramedics' mean response time this year is still 6.7 minutes (or higher) when it is actually less than 6.7 minutes.
Consequences: The city manager may take action to decrease paramedic response times when such action is unnecessary. This could result in considerable expense for the city, as well as some disgruntled paramedics.

48. Probabilities of Type I and Type II Errors...
Probability of Type I Error (rejecting Ho when null is really true): α, your significance level for the hypothesis test.
Probability of Type II Error (failing to reject Ho when alternative is really true): β. Very complicated to calculate. Beyond scope of this course.

49. Power of a Test...
Power: Probability that a test will reject Ho when Ha is true
Think of power as making the correct decision, not making an error, not making a mistake
High level of power is a good thing
Power = 1 – β (remember β is probability of making a type II error); so ‘power’ and β are complimentary
Just fyi. Know basics of power only. Really good examples of what power is on page 364 and 365 of text. Read it.

50. Power of a Test...
How can we increase power (making the correct decision)?
Increase α
Increase n
Decrease standard deviation (same effect as increasing the sample size, n)

51. Comparing Proportions from Two Populations: Hypothesis Testing
Ho: p1 = p2
Ha: p1 ≠ or > or < p2
We must first find the combined proportion 𝑝 𝑐 of successes in both samples combined
𝑝 𝑐 = 𝑐𝑜𝑢𝑛𝑡_𝑜𝑓_𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑒𝑠_𝑖𝑛_𝑏𝑜𝑡ℎ_𝑠𝑎𝑚𝑝𝑙𝑒𝑠_𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑 𝑐𝑜𝑢𝑛𝑡_𝑜𝑓_𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙𝑠_𝑖𝑛_𝑏𝑜𝑡ℎ_𝑠𝑎𝑚𝑝𝑙𝑒𝑠_𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑
= 𝑥 1 + 𝑥 2 𝑛 1 + 𝑛 2
Minitab will calculate p hat c automatically for you. Also acceptable to write p1 – p2 = 0 and p1 – p2 > or < or not = 0; move all over to left side.

52. Two Proportion Hypothesis Testing
Ho: p1 = p2
Ha: p1 ≠ or > or < p2
Minitab will calculate this for us; no need to memorize

53. Two Proportion Hypothesis Testing Conditions...
SRS – Each of the two samples must be SRSs from their respective populations or they must each be randomized experiments
Normality – Each of the following are all ≥ 10
(n1)( 𝑝 c)
(n1)(1 – 𝑝 c)
(n2)( 𝑝 c)
(n2)(1 – 𝑝 c)
You need to know how to calculate p hat combined so you can check above conditions.

54. Two Proportion Hypothesis Testing Conditions...
Independence
Each of the populations must be at least (10) times each of the corresponding sample sizes; and one sample does not influence the other

55. Confidence Interval for 𝑝 1 – 𝑝 2
To study the long-term effects of preschool programs for poor children, a research foundation has followed two groups of Michigan children since early childhood. A control group of 61 children represents Population 1, poor children with no pre-school. Another group of 62 from the same area and similar backgrounds attended pre-school as 3- and 4-year-olds represents Population 2, poor children who attend pre-school. Sizes are n1 = 61 and n2 = 62. One response variable of interest is the need for social services as adults. In the past ten years, 38 of the preschool sample and 49 of the control sample have needed social services (mainly welfare). Carry out an hypothesis test to determine if there is significant evidence that pre-school reduces or increases the later need for social services?

56. n pre-school = 62 nno pre-school = of pre-school needed social services; 49 of no pre-school needed social services
State null and alternative hypothesis Ho: pno pre-school = ppre-school Ha: pno pre-school ≠ ppre-school Conditions Randomization, Normality/Large Sample, Independence
Didn’t say SRS about either ; may not be able to generalize; normality n * p hat combined >= 10 and n (1 – p hat combined) >=10 for EACH n (n1 and n2); 4 conditions must be checked; independence... Pop must be at least 10 times sample size and one sample students does not influence any other child. NOTICE: n’s do not have to =; ok to be different n’s

57. Ho: pno pre-school = ppre-school Ha: pno pre-school ≠ ppre-school
Minitab to calculate test statistic, p-value, etc. Two Sample, Proportion, Options & Data
P-val of about 2%

58. Ho: pno pre-school = ppre-school Ha: pno pre-school ≠ ppre-school
Interpretation: Reject null hypothesis. At a significance level of 5% (α = 0.05), and a p-value of approximately 0.02 there is sufficient evidence to show that p no pre-school ≠ p pre-school
Always reject or fail to reject the null; then always address the alternative with explanation.

59. Fear of Crime...
The elderly fear crime more than younger people, even though they are less likely to be victims of crime. One of the few studies that looked at older blacks recruited random samples of 56 black women and 63 black men over the age of 65 from Atlantic City, New Jersey. Of the women, 27 said they “felt vulnerable” to crime; 46 of the men said this.
What proportion of women in the sample feel vulnerable? Of men? (Note: Men are victims of crime more often than women, so we expect a higher proportion of men to feel vulnerable.)
P hat for women 27/56 =
P hat for men 46/63 =
These are sample statistics. Is the variability just due to sampling variability? Or are the % really different?

60. Fear of Crime...
Test the hypothesis that the true, unknown population proportion of elderly black males who feel vulnerable is higher than that of elderly black women who feel vulnerable. Hypothesis, Conditions, Computations, Interpretation
Null: P (males) = P (females); alternative p (males) > p (females) OR CAN THROW ALL OVER TO LEFT SIDE; either ok. CONDITIONS: SRS, large sample/normality (check both samples and use combined p hat; all must be >= 10; Independence (pop at least 10 times sample size AND assume one doesn’t influence any other); COMPUTATIONS: Use MiniTab, don’t choose to use pooled calculations of proportions; this was originally to make manual calculations easier; but it is generally less accurate. P-val approx which is less than any reasonable alpha. So we reject null hypothesis. With a p-value of 0.02 and an alpha of 5% , we conclude that true, unknown population parameter, p, of elderly black males who feel vulnerable is higher than that of elderly black women who feel vulnerable.

61. Cholesterol & Heart Attacks...
High levels of cholesterol in the blood are associated with higher risk of heart attacks. Will using a drug to lower blood cholesterol reduce heart attacks? The Helsinki Heart Study looked at this question. Middle-aged men were assigned at random to one of two treatments: 2051 men took the drug gemfibrozil to reduce their cholesterol levels, and a control group of 2030 men took a placebo. During the next five years, 56 men in the gemfibrozil group and 84 men in the placebo group had heart attacks.
Is the apparent benefit of gemfibrozil statistically significant?
Again, notice that the n’s do not =; that’s ok. Go for it; start to finish with a partner; 10 mins.

62. Ho: pgemfibrozil = pplacebo Ha: pgemfibrozil < pplacebo
We want to use this comparative randomized experiment to draw conclusions about p1, the proportion of middle-aged men who would suffer heart attacks after taking gemfibrozil, and p2, the proportion of middle-aged men who would suffer heart attacks if they only took a placebo. We hope to show that gemfibrozil reduces heart attacks, so we have a one-sided alternative.
Note: you could also state as
Ho: pgemfibrozil – pplacebo = 0 Ha: pgemfibrozil – pplacebo < 0
P-val = ; reject null; state alpha, p-val, context; evidence that use of gemfibrozil does appear to reduce the proportion of middle aged men who suffer heart attacks.

63. A Civil Action
The movie A Civil Action tells the story of a major legal battle that took place in the small town of Woburn, Massachusetts. A town well that supplied water to East Woburn residents was contaminated by industrial chemicals. During the period that residents drank water from this well, a sample of 414 births showed 16 birth defects. On the west side of Woburn, a sample of 228 babies born during the same time period revealed 3 with birth defects. The plaintiffs suing the companies responsible for the contamination claimed that these data show that the rate of birth defects was significantly higher in East Woburn, where the contaminated well water was in use.
Assume all conditions have been checked and met. How strong is the evidence supporting this claim? What should the judge for this case conclude?

64. 𝑝 East = 16/414 = 𝑝 West = 3/228 =
Is the rate of birth defects in East Woburn higher than in West Woburn?
Ho: pEast = pWest or pEast – pWest = 0
Ha: pEast > pWest or pEast – pWest > 0
Is the difference 𝑝 East – 𝑝 West , – = statistically significant? (remember, these are just 𝑝 s; don’t determine that 2% is within rejection zone! This is NOT a p-value; you must actually do the test to reach a p-value).

65. 𝑝 East = 16/414 = 𝑝 West = 3/228 =
Ho: pEast = pWest or pEast – pWest = 0 Ha: pEast > pWest or pEast – pWest > 0 p-value = 0.034
Courts/judges tend to use 5% alpha/significance level; what if alpha had been 1%?

66. Seat Belt Use...
The proportion of drivers who use seat belts depends on things like age (young people are more likely to go unbelted) and gender (women are more likely to buckle up). It also depends on local law. Here are data from observing random samples of female Hispanic drivers in two cities:

67. Seat Belt Use...
Comparing local law suggests that a larger proportion of drivers wear seat belts in New York than in Boston. Do the data give good evidence that this is true for female Hispanic drivers? Justify your answer. Assume all conditions have been checked and met.

68. Ho: pNY = pB Ha: pNY > pB p-value = 0. 000000253 Reject Ho
Ho: pNY = pB Ha: pNY > pB p-value = Reject Ho. With a p-value of ≈ 0, there is strong evidence at any reasonable α that a smaller proportion of female Hispanic drivers wear seat belts in Boston than in New York.

69. We check conditions for a reason...
If conditions are not satisfied, our results may not be accurate, reliable, trustworthy, etc.