A random sample of 80 youths and a second random sample of 120 adults showed that 18 of the youths and 10 of the adults had been ticketed for careless driving. Use a 1% level of significance to test the claim that youth have a higher proportion of careless drivers than adults do. state the null and alternative hypothesis
You will need to use a formula for a binomial proportion two-sample z-test.
Proportion of adults = 10/120 = .083
Proportion of youths = 18/80 = .225
Use these proportions in the formula.
Null:
Ho: pY = pA (A = adults; Y = youths)
Alternative:
Ha: pY > pA
This will be a one-tailed test at .01 level of significance. (Use a z-table to determine the critical or cutoff value to reject the null.) If the test statistic calculated from the formula exceeds the critical value from the table, reject the null and conclude pY > pA. If the test statistic does not exceed the critical value from the table, do not reject the null (there is no difference).
I hope this brief explanation will help get you started.
(b) How many would you expect to be taller than 160 cm?
To test the claim that youth have a higher proportion of careless drivers than adults, we can use a hypothesis test.
Null Hypothesis (H0): The proportion of careless drivers in the youth population is equal to or less than the proportion of careless drivers in the adult population.
Alternative Hypothesis (Ha): The proportion of careless drivers in the youth population is higher than the proportion of careless drivers in the adult population.
Mathematically, we can represent the hypotheses as follows:
H0: p1 ≤ p2
Ha: p1 > p2
Where p1 is the proportion of youths ticketed for careless driving, and p2 is the proportion of adults ticketed for careless driving.
Now, to determine if there is evidence to suggest that youth have a higher proportion of careless drivers, we need to calculate the test statistic and compare it to the critical value.
Let's define:
p̂1 = Proportion of youths ticketed for careless driving = 18/80
p̂2 = Proportion of adults ticketed for careless driving = 10/120
To calculate the test statistic, we use the formula for the difference in sample proportions:
z = (p̂1 - p̂2) / √[(p*(1-p)*(1/n1 + 1/n2)]
Where p = (x1 + x2) / (n1 + n2)
x1 = Number of youths ticketed for careless driving
x2 = Number of adults ticketed for careless driving
n1 = Sample size of youths
n2 = Sample size of adults
Now we can substitute the values into the formula and calculate the test statistic.