Question

A proponent of a new proposition on a ballot wants to know whether the proposition is likely to pass. The proposition will pass if it gets more than​ 50% of the votes. Suppose a poll is​ taken, and 560 out of 1000 randomly selected people support the proposition. Should the proponent use a hypothesis test or a confidence interval to answer this​ question? Explain. If it is a hypothesis​ test, state the hypotheses and find the test​ statistic, p-value, and conclusion. Use a 5 ​% significance level. If a confidence interval is​ appropriate, find the approximate 90 ​% confidence interval. In both​ cases, assume that the necessary conditions have been met. Should the proponent use a hypothesis test or a confidence​ interval? (If both a hypothesis test and a confidence interval could be​ used, choose the simpler​ one.) A. The proponent should use a hypothesis test because the proponent wants to know whether or not the proposition will pass. A confidence interval would be useless. B. The proponent should use a hypothesis test because the proponent wants to know whether or not the proposition will pass.​ However, the proponent could also use a confidence interval. C. The proponent should use a confidence interval because the proponent wants to know the proportion of the population who will vote for the proposition. A hypothesis test would be impossible. D. The proponent should use a confidence interval because the proponent wants to know the proportion of the population who will vote for the proposition.​ However, the proponent could also use a hypothesis test. Determine the null and alternative hypotheses for the hypothesis test. Let p denote the population proportion of voters in favor of the proposition. Select the correct choice below​ and, if​ necessary, fill in the answer boxes within your choice. ​(Type integers or decimals. Do not​ round.) A. Upper H 0 ​: pless than nothingUpper H Subscript a ​: pgreater than nothing B. Upper H 0 ​: pequals nothingUpper H Subscript a ​: pgreater than nothing C. Upper H 0 ​: pequals nothingUpper H Subscript a ​: pnot equals nothing D. Upper H 0 ​: pequals nothingUpper H Subscript a ​: pless than nothing E. Upper H 0 ​: pgreater than nothingUpper H Subscript a ​: pless than nothing F. A hypothesis test is not the most appropriate approach. The proponent should use a confidence interval. Find the test statistic for the hypothesis test. Select the correct choice below​ and, if​ necessary, fill in the answer box within your choice. A. zequals nothing ​(Round to two decimal places as​ needed.) B. A hypothesis test is not the most appropriate approach. The proponent should use a confidence interval. Find the​ p-value. Select the correct choice below​ and, if​ necessary, fill in the answer box within your choice. A. ​p-valueequals nothing ​(Round to three decimal places as​ needed.) B. A hypothesis test is not the most appropriate approach. The proponent should use a confidence interval. Determine the proper conclusion to the hypothesis test. Choose the correct answer below. A. Do not reject Upper H 0 . There is not enough evidence to conclude that the proposition will pass. B. Reject Upper H 0 . There is not enough evidence to conclude that the proposition will pass. C. Do not reject Upper H 0 . There is enough evidence to conclude that the proposition will pass. D. Reject Upper H 0 . There is enough evidence to conclude that the proposition will pass. E. A hypothesis test is not the most appropriate approach. The proponent should use a confidence interval. Construct an approximate 90 ​% confidence interval for the population proportion p. Select the correct choice below​ and, if​ necessary, fill in the answer boxes within your choice. A. left parenthesis nothing comma nothing right parenthesis ​(Round to two decimal places as​ needed.) B. A confidence interval is not the most appropriate approach. The proponent should use a hypothesis test. Click to select and enter your answer(s).

Answer 1

Solution:-

B) The proponent should use a hypothesis test because the proponent wants to know whether or not the proposition will pass. However, the proponent could also use a confidence interval.

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P < 0.50
Alternative hypothesis: P > 0.50

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method, shown in the next section, is a one-sample z-test.

Analyze sample data. Using sample data, we calculate the standard deviation (S.D) and compute the z-score test statistic (z).

S.D = sqrt[ P * ( 1 - P ) / n ]

S.D = 0.01581
z = (p - P) / S.D

z = 3.795

where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and n is the sample size.

Since we have a one-tailed test, the P-value is the probability that the z-score is greater than 3.795.

Thus, the P-value = less than 0.0001.

Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we cannot accept the null hypothesis.

Reject Upper H 0 . There is enough evidence to conclude that the proposition will pass.

90 % confidence interval for the population proportion is C.I = ( 0.534, 0.586).

C.I = 0.56 + 1.645 × 0.01581

C.I = 0.56 + 0.02601

C.I = ( 0.534, 0.586)