A test of 60 youths and 140 adults showed that 22 of the youths and 10 of the adults were careless drivers. Use a 1% level of significance to test the claim that the youth percentage of careless drivers is higher than the adult percentage. Next find the 99% confidence interval for this problem.
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To test the claim that the youth percentage of careless drivers is higher than the adult percentage, we can use the chi-square test for comparing two proportions.
First, let's calculate the observed proportions of careless drivers in the youth and adult groups:
Proportion of careless drivers among youths = 22/60 = 0.3667
Proportion of careless drivers among adults = 10/140 = 0.0714
Now, let's calculate the expected proportions assuming that there is no difference between the two groups:
Expected proportion of careless drivers among youths = (total careless drivers / total sample size) = (22 + 10) / (60 + 140) = 0.1333
Now, let's set up the null and alternative hypotheses:
Null hypothesis (H0): The youth percentage of careless drivers is not higher than the adult percentage.
Alternative hypothesis (Ha): The youth percentage of careless drivers is higher than the adult percentage.
Since we want to test if the youth percentage is higher, this is a one-tailed test.
Next, we need to calculate the chi-square test statistic and compare it to the critical chi-square value at a 1% significance level. The chi-square test statistic can be calculated using the formula:
chi-square = (observed1 - expected1)^2 / expected1 + (observed2 - expected2)^2 / expected2
Substituting the values:
chi-square = (0.3667 - 0.1333)^2 / 0.1333 + (0.0714 - 0.1333)^2 / 0.1333
Calculating this expression gives us the chi-square test statistic.
Next, we need to find the critical chi-square value at a 1% level of significance with 1 degree of freedom (since we have 1 categorical variable: youths vs. adults). You can use statistical tables or a chi-square calculator to find this value.
If the calculated chi-square test statistic is greater than the critical chi-square value, we reject the null hypothesis. If it is less than or equal to the critical value, we fail to reject the null hypothesis.
Now, to find the 99% confidence interval for the difference in proportions, we can use the formula:
Confidence Interval = (observed proportion - margin of error, observed proportion + margin of error)
The margin of error can be calculated using the formula:
Margin of Error = critical value * standard error
The standard error can be calculated using the formula:
Standard Error = sqrt((p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2))
where p1 and p2 are the observed proportions and n1 and n2 are the respective sample sizes.
Substituting the values, we can calculate the standard error and then find the margin of error. Finally, we can construct the confidence interval using the observed proportion and the margin of error.
Note: The exact calculations and critical values may vary depending on the statistical software or calculator you are using.