Thank you for visiting Two different rifles are tested at the shooting range 190 shots were fired from the first rifle and the target was hit 158 times 140. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
Answer:
(-0.02 ; 0.16)
Step-by-step explanation:
n1 = 190 ; n2 = 140
hits 1, x1 = 158 ; hits 2, x2 = 107
P1 = x1/n1 = 158 / 190 = 0.8316
P2 = x2/n2 = 107 / 140 = 0.7643
Zcritical at 95% = 1.96
Confidence interval :
(P1 - P2) ± Zcritical * standard error
Standard Error = sqrt[(p1(1-p1))/n1 + (p2(1-p2))/n2]
S.E = sqrt((0.8316(0.1684))/190 + (0.7643(0.2357))/140) = 0.0449868
(0.8316 - 0.7643) ± (1.96 * 0.0449868)
0.0673 ± 0.0881741
Lower boundary = 0.0673 - 0.0881741 = −0.020874
Upper boundary = 0.0673 - 0.0881741 = 0.1554741
(-0.02 ; 0.16)
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Final answer:
The 95% confidence interval for the difference in accuracy between the two rifles, computed using the formula for the confidence interval of the difference of proportions, is (-0.0067, 0.1413).
Explanation:
The subject is the comparison of the accuracy rates of two different rifles. To create a 95% confidence interval for the difference in the accuracy rates, we first tackle each rifle separately. For the first rifle, the hit rate is 158/190 = 0.8316. For the second, the hit rate is 107/140 = 0.7643.
We subtract these proportions to get the point estimate of the difference, which is 0.8316 - 0.7643 = 0.0673.
Next, we calculate the standard error of the point estimate using the formula sqrt[(p1(1-p1)/n1) + (p2(1-p2)/n2)], where p1 and p2 are the hit rates for each rifle and n1 and n2 are the number of shots each fired. After calculation, the standard error is approximately 0.038.
For a 95% confidence interval, the z-score is about 1.96. The margin of error is the product of the z-score and the standard error, which is approximately 0.074.
The 95% confidence interval is then (0.0673 - 0.074, 0.0673 + 0.074) = (-0.0067, 0.1413). Thus, we estimate with 95% confidence that the true difference in the rifles' accuracy lies within this range.
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