SOLUTION: study was conducted to investigate the effectiveness of bicycle safety helmets in preventing head injury. The data consist of a random sample of 793 persons who were involved in b

Question 1172241: study was conducted to investigate the effectiveness of bicycle safety helmets in preventing
head injury. The data consist of a random sample of 793 persons who were involved in bicycle
accidents during an one‐year period (Table 1).
Table 1: Head Injury data
Head injury Wearing helmet - Yes Wearing helmet - No
Yes 17 218
No 130 428
a. Compute and compare the proportions of head injury for the group with helmets
versus the group without helmets. What would be your conclusion?
b. Use appropriate statistical techniques to support your conclusion.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
how to deal with this type of problem is covered in the following reference.

https://www.dummies.com/education/math/statistics/how-to-compare-two-population-proportions/

your overall sample size is 793
this is divided into two groups.
the first group wore helmets.
the second group didn't wear helmets.

the proportion of people in the first group who had head injuries (the ones who wore helmets) is 17/235.
the proportion of people in the second group who had head injuries (the ones who didn't wear helmets) is 130/558.
the proportion of people overall who had head injuries is 147/793.

calculate the overall standard error.
that would be equal to the square root of overall proportion of people who had head injuries times the overall proportion of people who didn't have head injuries times the sum of the reciprocal of the number of people who were in the group that wore helmets plus the reciprocal of the number of people who were in the group that didn't wear helmets.

algebraically, you would show this as.

s = sqrt(p0 * (1 - p0) * (1/n1 + 1/n2))
p0 is equal to 147/793 *** this is the overall proportion of people who had head injuries.
1 - p0 is equal to 1 - 147/793 which is equal to 646/793 *** this is the overall proportion of people who didn't have head injuries.
1/n1 is equal to 1/235 *** this is the reciprocal of the number of people in the group that wore helmets.
1/n2 is equal to 1/558 *** this is the reciprocal of the number of people in the group that didn't wear helmets.
the formula becomes:
s = sqrt(147/793 * 646/793 * (1/235 + 1/558)) = .0302195019

your z-score formula is:
z = (x1 - x2) / s
z is the z-score.
x1 is the proportion of people who wore helmets and had head injuries.
x2 is the proportion of people who did not wear helmets and had head injuries.
s is the standard error we just completed calculating.
the formula becomes:
z = (17/235 - 130/558) / .0302195019 = -5.315590086

note that the reference assumes (x1 - x2) is being compared against 0.
technically, the z-score formula should be:
z = ((x1-x2)-0)/s
however, anything that you subtract 0 from is equal to that same anything, so not showing (x1 - x2) - 0 and showing (x1 - x2) by itself winds up being the same thing.
this is why i showed (x1 - x2) rather than (x1 - x2) - 0.
they're equivalent.

the critical z-score for a two tailed confidence level of 99% would be plus or minus 2.575829303.
a z-score of 5.3155 is well beyond the critical z-score threshold.
this means that it is highly unlikely that such a z-score could be a result of random variations in the mean of samples of the sizes indicated.
the resulting conclusion is that there is a significant difference between the proportion of people who had head injuries and wore helmets versus the proportion of people who had head injuries and didn't wear helmets.

the conclusion.
you're a lot better off wearing a helmet than not when riding a bicycle.

read the reference to understand how this study was conducted.
in the reference, they mention p-hat.
that's a p with a small ^ on top of it.
this sometimes referred to as p-hat
that's the proportion of the sample.
p-hat1 is what i showed as p1.
p-hat2 is what i showed as p2.
p-hat overall is what i showed as p0.

let me know if you have any questions with the reference and/or what i did with it.

theo

RELATED QUESTIONS
NHTSA new car crash tests. Refer to the National Highway Traffic Safety Administration... (answered by CPhill)

The more snowboarders wear helmets the less likely they are to sustain head injuries. In... (answered by jim_thompson5910)

A study was conducted to investigate the effectiveness of hypnotism in reducing pain.... (answered by Boreal)

Consider a study conducted to evaluate the effectiveness of a new treatment to stop... (answered by ewatrrr)

A small pilot study is conducted to investigate the effect of a nutritional supplement on (answered by CPhill)

In a sample of 100 Republicans, 72 favored the president’s anti - drug program; while (answered by ikleyn)

In a large U.S. city a study was conducted to determine the typical commute time from... (answered by stanbon)

2. N Hill et al conducted a clinical study to compare the standard treatment for head... (answered by stanbon)

A study was conducted to investigate the relationship between the cost, y (in tens of... (answered by stanbon)