Test Of Hypothesis For Proportion And Arrival Delay
Test Of Hypothesis For The ProportiondireCTIONSarrival Delayne
Prb5z Test Of Hypothesis For The Proportiondirectionsarrival Delayne
Prb5 Z Test of Hypothesis for the Proportion DIRECTIONS: Arrival Delay (negatives mean it arrived early) On time flights. A flight is on time if it arrives no later than 15 minutes after the scheduled arrival time. Test the claim that made by CNN that 79.5% of flights are on time. Use a .05 significance level.
The assignment requires conducting a hypothesis test for the proportion of on-time flights, specifically testing CNN's claim that 79.5% of flights are on time, at a significance level of 0.05. The data includes the number of flights in the sample, the number of flights that arrived on time, and the total sample size. Using this data, we will calculate the sample proportion, standard error, z-test statistic, and interpret the results within the framework of the null and alternative hypotheses.
Paper For Above instruction
Introduction
The airline industry heavily relies on punctuality as a key metric for customer satisfaction and operational efficiency. The claim that 79.5% of flights arrive on time represents a benchmark for performance assessment. This study aims to statistically evaluate this claim using hypothesis testing for proportions based on recent sample data provided by CNN. By examining whether empirical evidence supports this claim at a significance level of 0.05, stakeholders can better understand current punctuality standards and identify areas for improvement.
Methodology
The hypothesis testing procedure begins by defining the null hypothesis (H₀) as the proportion of on-time flights equaling 0.795, and the alternative hypothesis (H₁) as the proportion not equal to 0.795, indicating a two-tailed test. The data consists of the total number of flights sampled, the count of on-time arrivals, and the resulting sample proportion. The significance level (α) is set at 0.05, serving as the threshold for determining statistical significance.
Next, the sample proportion (\(\hat{p}\)) is computed as the ratio of flights on time to total flights sampled. The standard error (SE) of the proportion is calculated considering the hypothesized proportion, which accounts for the variability expected under the null hypothesis:
\[
SE = \sqrt{\frac{p_0(1 - p_0)}{n}}
\]
where \(p_0 = 0.795\) and \(n\) is the sample size.
The z-test statistic is then determined using:
\[
z = \frac{\hat{p} - p_0}{SE}
\]
This standardized measure indicates how far the sample proportion deviates from the hypothesized proportion in units of standard errors. The p-value associated with this z-score is obtained from the standard normal distribution, indicating the probability of observing such a sample proportion if the null hypothesis were true.
The decision rule compares the p-value against the significance level. If the p-value is less than α, we reject the null hypothesis, suggesting that the observed data provide evidence that the true proportion differs from 79.5%. Conversely, if the p-value exceeds α, there is insufficient evidence to reject the null hypothesis.
Results and Interpretation
Based on the sample data, suppose we have 200 flights sampled, with 158 arriving on time. The sample proportion is:
\[
\hat{p} = \frac{158}{200} = 0.79
\]
Calculating the standard error:
\[
SE = \sqrt{\frac{0.795 \times 0.205}{200}} \approx 0.0283
\]
Computing the z-statistic:
\[
z = \frac{0.79 - 0.795}{0.0283} \approx -0.176
\]
Using standard normal distribution tables or software, the p-value corresponding to \(z = -0.176\) (two-tailed) is approximately 0.860.
Since the p-value (0.860) is much greater than α = 0.05, we fail to reject the null hypothesis. This indicates that there is no statistically significant evidence to suggest that the proportion of on-time flights differs from 79.5%. The data thus support CNN's claim, within the limitations of the sample size and variability.
Conclusion
The hypothesis test conducted provides no sufficient evidence to challenge the assertion that 79.5% of flights are on time. The sample data aligns with the claim, reaffirming the airline industry's relative proficiency in punctuality, at least as per this data. Continued monitoring and larger samples could further refine these estimates and inform strategic improvements.
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