We would like to determine p, the probability that a tossed coin shows heads. Th
ID: 3159918 • Letter: W
Question
We would like to determine p, the probability that a tossed coin shows heads. The coin was tossed 200 times and it showed heads 117 times.
1. Construct a 90% confidence interval for p.
2. We wish to perform a hypothesis test at the 5% significance level that the coin is biased in favor of heads. Make sure to state the null and alternative hypotheses, state whether it is a left-, right- or two-tailed test, show all necessary steps to arrive at decision, and state your final conclusion in the context of this problem.
Explanation / Answer
1.
Note that
p^ = point estimate of the population proportion = x / n = 0.585
Also, we get the standard error of p, sp:
sp = sqrt[p^ (1 - p^) / n] = 0.034840709
Now, for the critical z,
alpha/2 = 0.05
Thus, z(alpha/2) = 1.644853627
Thus,
Margin of error = z(alpha/2)*sp = 0.057307866
lower bound = p^ - z(alpha/2) * sp = 0.527692134
upper bound = p^ + z(alpha/2) * sp = 0.642307866
Thus, the confidence interval is
( 0.527692134 , 0.642307866 ) [ANSWER]
********************************
2.Formulating the null and alternatuve hypotheses,
Ho: p <= 0.5
Ha: p > 0.5 [HYPOTHESES]
******************
Hence, this is a right tailed test.
******************
As we see, the hypothesized po = 0.5
Getting the point estimate of p, p^,
p^ = x / n = 0.585
Getting the standard error of p^, sp,
sp = sqrt[po (1 - po)/n] = 0.035355339
Getting the z statistic,
z = (p^ - po)/sp = 2.404163056
As this is a 1 tailed test, then, getting the p value,
p = 0.008104771
As P < 0.05, we REJECT THE NULL HYPOTHESIS.
Hence, there is significant evidence that the coin is biased in favor of heads. [CONCLUSION]
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.