Prb5 Z Test Of Hypothesis For The Proportion Directions ✓ Solved

Prb5 Z Test of Hypothesis for the Proportion DIRECTIONS:

A flight is on time if it arrives no later than 15 minutes after the scheduled arrival time. Test the claim made by CNN that 79.5% of flights are on time. Use a .05 significance level.

Paper For Above Instructions

The purpose of this paper is to conduct a hypothesis test to determine if the claim made by CNN that 79.5% of flights are on time is valid. A flight is deemed on time if it arrives no later than 15 minutes after the scheduled arrival time. We will utilize a significance level of 0.05 for our analysis.

Step 1: Formulating the Hypotheses

The first step in conducting a hypothesis test is to formulate the null and alternative hypotheses:

  • Null Hypothesis (H0): p = 0.795 (The proportion of on-time flights is 79.5%.)
  • Alternative Hypothesis (H1): p ≠ 0.795 (The proportion of on-time flights is not 79.5%.)

Step 2: Collecting Sample Data

The next step is to collect data regarding the actual flight arrivals. For this example, let's assume we have a sample size of 200 flights. Out of those, we observe that 160 flights arrived on time.

  • Sample Size (n): 200
  • Number of On-time Flights (x): 160

Step 3: Calculate the Sample Proportion

We calculate the sample proportion (p̂) of on-time flights:

p̂ = x/n = 160/200 = 0.80

Step 4: Compute the Standard Error

The standard error (SE) of the sample proportion can be calculated using the formula:

SE = sqrt[(p(1-p))/n] where p is the population proportion (0.795).

SE = sqrt[(0.795 (1 - 0.795)) / 200] = sqrt[(0.795 0.205) / 200] ≈ 0.0355.

Step 5: Calculate the Z-Test Statistic

The Z-test statistic is calculated using the formula:

Z = (p̂ - p) / SE

Substituting in our values:

Z = (0.80 - 0.795) / 0.0355 ≈ 0.1414.

Step 6: Determine the Critical Value

For a two-tailed test at a significance level of 0.05, the critical Z-values are approximately ±1.96.

Step 7: Make a Decision

We will reject the null hypothesis if the calculated Z falls outside the critical values. In our case, the calculated Z of 0.1414 is within the range of -1.96 to 1.96.

Since 0.1414 is less than 1.96 and greater than -1.96, we fail to reject the null hypothesis.

Step 8: Interpretation of Results

The results of the hypothesis test indicate that we do not have sufficient evidence to conclude that the proportion of on-time flights differs from 79.5%. Therefore, we cannot refute the claim made by CNN.

Conclusion

This analysis employed a Z-test for proportions and demonstrated the steps involved in conducting a hypothesis test. By properly formulating the null and alternative hypotheses, calculating the required statistics, and interpreting the results, we provided a clear evaluation of the claim regarding the timeliness of flights.

References

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