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Analyze the provided case studies involving hypothesis testing for election results and customer payment periods, focusing on the application of one-sample hypothesis tests, significance levels, and decision-making based on test statistics and critical values. Your task is to interpret the data, formulate hypotheses, perform the relevant statistical tests, and draw conclusions about the outcomes.

Specifically, examine the election case involving the exit poll with 765 voters where George W. Bush obtained 407 votes. Determine whether Bush is statistically likely to have received more than 50% of votes at a 0.10 significance level, performing a right-tailed Z-test for proportions. Interpret the test statistic, compare it with the critical value (1.28), and decide whether to reject or fail to reject the null hypothesis that Bush's proportion is exactly 50%.

Similarly, consider the SpeedX case where a sample of 220 customers shows an average payment time of 21.63 days with a standard deviation of 5.84 days. Test whether including stamped self-addressed envelopes reduces the average payment time to 22 days at a 0.10 significance level, using a left-tailed Z-test for the population mean. Compute the test statistic, compare it with the critical value (-1.28), and conclude whether there is sufficient statistical evidence to support the claim.

Sample Paper For Above instruction

Title: Hypothesis Testing of Election Results and Payment Time Reduction at Significance Level 0.10

Introduction

Hypothesis testing is a critical statistical tool used in decision-making processes across various fields, including political elections and business operations. It involves setting up a null hypothesis (H0) and an alternative hypothesis (H1), calculating a test statistic based on the sample data, and comparing it to a critical value at a specified significance level (α). The resulting decision guides whether to accept or reject the null hypothesis. This paper applies one-sample hypothesis testing to two case studies: the 2000 US presidential election exit poll and a payment time reduction study at SpeedX, illustrating the practical use of statistical inference in real-world scenarios.

Election Results Hypothesis Testing

The election case involves a survey of 765 voters where George W. Bush received 407 votes, and the network predicts a candidate wins if they secure more than 50% of votes. The null hypothesis (H0: p = 0.50) posits that Bush's proportion of votes is exactly 50%. The alternative hypothesis (H1: p > 0.50) suggests Bush's vote share exceeds 50%. The significance level chosen is 0.10, and the critical value for a right-tailed Z-test is 1.28.

The sample proportion (p̂) is calculated as 407/765 ≈ 0.531. The test statistic is computed using the formula:

z = (p̂ - p0) / √(p0(1 - p0)/n) = (0.531 - 0.50) / √(0.50*0.50/765) ≈ 1.77

Since the test statistic (1.77) exceeds the critical value (1.28), the null hypothesis is rejected. This statistical evidence supports the conclusion that Bush's proportion of votes is significantly greater than 50% at the 0.10 significance level, justifying the network's early declaration of victory.

Payment Time Reduction Hypothesis Testing

The SpeedX case assesses whether sending stamped self-addressed envelopes decreases the average payment time. The sample data includes 220 customers with a mean payment of 21.63 days, standard deviation 5.84 days, and hypothesized mean (H0) of 22 days. The alternative hypothesis (H1: μ

The test statistic is calculated as:

z = (x̄ - μ0) / (σ / √n) = (21.63 - 22) / (5.84 / √220) ≈ -0.91

Since -0.91 > -1.28, the test statistic falls outside the rejection region, and we fail to reject the null hypothesis. Therefore, there is insufficient evidence at the 0.10 level to claim that the new mailing procedure reduces payment times by 2 days.

Conclusion

The hypothesis tests demonstrate the effectiveness of statistical inference in decision-making. In the election case, the data provide sufficient evidence to support that Bush received more than half of the votes, corroborating early media projections. Conversely, in the payment time study, the evidence does not support a significant reduction in payment duration, indicating that further analysis or larger samples may be necessary. Applying these robust methods enables organizations and entities to make informed, data-driven decisions with confidence.

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