Complete the following steps to interpret a 1-sample Poisson rate test. Key output includes the rate of occurrence, the confidence interval, and the p-value.

First, consider the sample rate, and then examine the confidence interval.

The sample rate of an event is an estimate of the population rate of that event. Because the sample rate is based on sample data and not on the entire population, it is unlikely that the sample rate equals the population rate of occurrence. To better estimate the population rate of occurrence, use the confidence interval.

The confidence interval provides a range of likely values for the population rate of occurrence. For example, a 95% confidence level indicates that if you take 100 random samples from the population, you could expect approximately 95 of the samples to produce intervals that contain the population rate. The confidence interval helps you assess the practical significance of your results. Use your specialized knowledge to determine whether the confidence interval includes values that have practical significance for your situation. If the interval is too wide to be useful, consider increasing your sample size. For more information, go to Ways to get a more precise confidence interval.

N | Total Occurrences | Sample Rate | 95% CI for λ |
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30 | 598 | 19.9333 | (18.3675, 21.5970) |

In these results, the estimate of the population rate of occurrence for the number of customer complaints per day is approximately 19.93. You can be 95% confident that the population rate of occurrence is between approximately 18.37 and 21.6.

To determine whether the difference between the population rate and the hypothesized rate is statistically significant, compare the p-value to the significance level. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference.

- P-value ≤ α: The difference between the rates is statistically significant (Reject H
_{0}) - If the p-value is less than or equal to the significance level, the decision is to reject the null hypothesis. You can conclude that the difference between the population rate and the hypothesized rate is statistically significant. Use your specialized knowledge to determine whether the difference is practically significant. For more information, go to Statistical and practical significance.
- P-value > α: The difference between the rates is not statistically significant (Fail to reject H
_{0}) - If the p-value is greater than the significance level, the decision is to fail to reject the null hypothesis. You do not have enough evidence to conclude that the difference between the population rate and the hypothesized rate is statistically significant. You should make sure that your test has enough power to detect a difference that is practically significant. For more information, go to Power and Sample Size for 1-Sample Poisson Rate.

λ: Poisson rate of Number of complaints |
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Exact method is used for this analysis. |

N | Total Occurrences | Sample Rate | 95% Lower Bound for λ |
---|---|---|---|

30 | 598 | 19.9333 | 18.6118 |

Null hypothesis | H₀: λ = 10 |
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Alternative hypothesis | H₁: λ > 10 |

P-Value |
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0.000 |

In these results, the null hypothesis states that the rate is 10 complaints per day. Because the p-value of 0.000 is less than the significance level of 0.05, the manager rejects the null hypothesis and concludes that the rate of complaints is greater than 10 per day.