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Interpret all statistics for 2-Sample Poisson Rate

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Find definitions and interpretation guidance for every statistic that is provided with the 2-sample Poisson rate analysis.

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Difference = rate1 - rate2

The difference is the unknown difference between the population rates that you want to estimate. Minitab indicates which population rate is subtracted from the other.

Observation length

Poisson processes count occurrences of a certain event or property on a specific observation range, which can represent things such as time, area, volume, and number of items. The observation length represents the magnitude, duration, or size of each observation range.

Interpretation

Minitab uses the observation length to convert the sample rate into a form that best suits your situation.

For example, if each sample observation counts the number of events in a year, a length of 1 represents a yearly rate of occurrence, and a length of 12 represents a monthly rate of occurrence.

Minitab uses total occurrences, the sample size (N), and the observation length to calculate the sample rate. For example, inspectors inspect the number of defects in boxes of towels from 2 assembly lines (A and B). A towel can have more than one defect, such as 1 tear and 2 pulls (3 defects). For assembly line A, each box contains 10 towels. The inspectors sample 50 total boxes and find a total of 112 defects. For assembly line B, each box contains 15 towels. The inspectors sample 50 total boxes and find a total of 132 defects.
  • For assembly line A, the total occurrences is 112 because the inspectors find 112 defects. It is 132 for assembly line B because the inspectors find 132 defects.
  • The sample size (N) is 50 for both assembly lines because the inspectors sample 50 boxes for both assembly lines.
  • To determine the number of defects per towel, the inspectors use an observation length of 10 for assembly line A because there are 10 towels per box. For assembly line B, the inspectors use an observation length of 15.
  • For assembly line A, the sample rate is (Total occurrences / N) / (observation length) = (112/50) / 10 = 0.224. For assembly line B, the sample rate is (132/50) / 15 = 0.176. So, on average, each towel from assembly line A has 0.244 defects, and each towel from assembly line B has 0.176 defects.
  • Because the inspectors enter an observation length that is different from 1, Minitab also calculates the sample mean. For assembly line A, the sample mean is (Total occurrences / N) = 112/50 = 2.24. For assembly line B, the sample mean is 132/50 = 2.64. The sample mean describes the average number of defects per box. However, because the boxes had different amounts of towels, the sample rate is a more helpful statistic.

Total Occurrences

The total occurrences is the number of times an event occurs in the sample.

Minitab uses total occurrences, the sample size (N), and the observation length to calculate the sample rate. For example, inspectors inspect the number of defects in boxes of towels from 2 assembly lines (A and B). A towel can have more than one defect, such as 1 tear and 2 pulls (3 defects). For assembly line A, each box contains 10 towels. The inspectors sample 50 total boxes and find a total of 112 defects. For assembly line B, each box contains 15 towels. The inspectors sample 50 total boxes and find a total of 132 defects.
  • For assembly line A, the total occurrences is 112 because the inspectors find 112 defects. It is 132 for assembly line B because the inspectors find 132 defects.
  • The sample size (N) is 50 for both assembly lines because the inspectors sample 50 boxes for both assembly lines.
  • To determine the number of defects per towel, the inspectors use an observation length of 10 for assembly line A because there are 10 towels per box. For assembly line B, the inspectors use an observation length of 15.
  • For assembly line A, the sample rate is (Total occurrences / N) / (observation length) = (112/50) / 10 = 0.224. For assembly line B, the sample rate is (132/50) / 15 = 0.176. So, on average, each towel from assembly line A has 0.244 defects, and each towel from assembly line B has 0.176 defects.
  • Because the inspectors enter an observation length that is different from 1, Minitab also calculates the sample mean. For assembly line A, the sample mean is (Total occurrences / N) = 112/50 = 2.24. For assembly line B, the sample mean is 132/50 = 2.64. The sample mean describes the average number of defects per box. However, because the boxes had different amounts of towels, the sample rate is a more helpful statistic.

N

The sample size (N) is the total number of observations in the sample.

Interpretation

The sample size affects the confidence interval, the power of the test, and the rate of occurrence.

Usually, a larger sample results in a narrower confidence interval. A larger sample size also gives the test more power to detect a difference. For more information, go to What is power?.

Minitab uses total occurrences, the sample size (N), and the observation length to calculate the sample rate. For example, inspectors inspect the number of defects in boxes of towels from 2 assembly lines (A and B). A towel can have more than one defect, such as 1 tear and 2 pulls (3 defects). For assembly line A, each box contains 10 towels. The inspectors sample 50 total boxes and find a total of 112 defects. For assembly line B, each box contains 15 towels. The inspectors sample 50 total boxes and find a total of 132 defects.
  • For assembly line A, the total occurrences is 112 because the inspectors find 112 defects. It is 132 for assembly line B because the inspectors find 132 defects.
  • The sample size (N) is 50 for both assembly lines because the inspectors sample 50 boxes for both assembly lines.
  • To determine the number of defects per towel, the inspectors use an observation length of 10 for assembly line A because there are 10 towels per box. For assembly line B, the inspectors use an observation length of 15.
  • For assembly line A, the sample rate is (Total occurrences / N) / (observation length) = (112/50) / 10 = 0.224. For assembly line B, the sample rate is (132/50) / 15 = 0.176. So, on average, each towel from assembly line A has 0.244 defects, and each towel from assembly line B has 0.176 defects.
  • Because the inspectors enter an observation length that is different from 1, Minitab also calculates the sample mean. For assembly line A, the sample mean is (Total occurrences / N) = 112/50 = 2.24. For assembly line B, the sample mean is 132/50 = 2.64. The sample mean describes the average number of defects per box. However, because the boxes had different amounts of towels, the sample rate is a more helpful statistic.

Sample rate

The sample rate of an event is the average number of times the event occurs per unit length of observation in the sample.

Interpretation

The sample rate of each sample is an estimate of the population rate of each sample.

Minitab uses total occurrences, the sample size (N), and the observation length to calculate the sample rate. For example, inspectors inspect the number of defects in boxes of towels from 2 assembly lines (A and B). A towel can have more than one defect, such as 1 tear and 2 pulls (3 defects). For assembly line A, each box contains 10 towels. The inspectors sample 50 total boxes and find a total of 112 defects. For assembly line B, each box contains 15 towels. The inspectors sample 50 total boxes and find a total of 132 defects.
  • For assembly line A, the total occurrences is 112 because the inspectors find 112 defects. It is 132 for assembly line B because the inspectors find 132 defects.
  • The sample size (N) is 50 for both assembly lines because the inspectors sample 50 boxes for both assembly lines.
  • To determine the number of defects per towel, the inspectors use an observation length of 10 for assembly line A because there are 10 towels per box. For assembly line B, the inspectors use an observation length of 15.
  • For assembly line A, the sample rate is (Total occurrences / N) / (observation length) = (112/50) / 10 = 0.224. For assembly line B, the sample rate is (132/50) / 15 = 0.176. So, on average, each towel from assembly line A has 0.244 defects, and each towel from assembly line B has 0.176 defects.
  • Because the inspectors enter an observation length that is different from 1, Minitab also calculates the sample mean. For assembly line A, the sample mean is (Total occurrences / N) = 112/50 = 2.24. For assembly line B, the sample mean is 132/50 = 2.64. The sample mean describes the average number of defects per box. However, because the boxes had different amounts of towels, the sample rate is a more helpful statistic.

Sample Mean

When the observed length is different from 1, Minitab displays the sample mean. The sample mean is the total number of occurrences divided by the sample size. However, because the observation length differs from 1, the sample rate will usually be more useful for your particular situation.

Estimated difference

The estimated difference is the difference between the rates of occurrence of the two samples.

Because the difference is based on sample data and not on the entire population, it is unlikely that the sample difference equals the population difference. To better estimate the population difference, use the confidence interval for the difference.

Confidence interval (CI) and bounds

The confidence interval provides a range of likely values for the population difference. Because samples are random, two samples from a population are unlikely to yield identical confidence intervals. But, if you repeated your sample many times, a certain percentage of the resulting confidence intervals or bounds would contain the unknown population difference. The percentage of these confidence intervals or bounds that contain the difference is the confidence level of the interval. 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 difference.

An upper bound defines a value that the population difference is likely to be less than. A lower bound defines a value that the population difference is likely to be greater than.

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.

Estimation for Difference

Estimated
Difference		95% CI for Difference
-7.7(-14.6768, -0.723175)

In these results, the estimate of the population rate of occurrence for the difference in customer visits for two post offices is −7.7. You can be 95% confident that the difference in population rates is between approximately −14.7 and −0.7.

Null hypothesis and alternative hypothesis

The test for difference displays the null and alternative hypotheses. The null and alternative hypotheses are two mutually exclusive statements about a population. A hypothesis test uses sample data to determine whether to reject the null hypothesis.
Null hypothesis
The null hypothesis states that a population parameter (such as the mean, the standard deviation, and so on) is equal to a hypothesized value. The null hypothesis is often an initial claim that is based on previous analyses or specialized knowledge.
Alternative hypothesis
The alternative hypothesis states that a population parameter is smaller, larger, or different from the hypothesized value in the null hypothesis. The alternative hypothesis is what you might believe to be true or hope to prove true.

In the output, the null and alternative hypotheses help you to verify that you entered the correct value for the test difference.

Z-Value

The Z-value is a test statistic for Z-tests that measures the difference between an observed statistic and its hypothesized population parameter in units of standard error.

Interpretation

You can compare the Z-value to critical values of the standard normal distribution to determine whether to reject the null hypothesis. However, using the p-value of the test to make the same determination is usually more practical and convenient.

To determine whether to reject the null hypothesis, compare the Z-value to your critical value. The critical value is Z1-α/2 for a two–sided test and Z1-α for a one–sided test. For a two-sided test, if the absolute value of the Z-value is greater than the critical value, you reject the null hypothesis. If the absolute value of the Z-value is less than the critical value, you fail to reject the null hypothesis. You can calculate the critical value in Minitab or find the critical value from a standard normal table in most statistics books. For more information, go to Using the inverse cumulative distribution function (ICDF) and click "Use the ICDF to calculate critical values".

The Z-value is used to calculate the p-value.

P-Value

The p-value is a probability that measures the evidence against the null hypothesis. A smaller p-value provides stronger evidence against the null hypothesis.

Interpretation

Use the p-value to determine whether the difference in population rates of occurrence is statistically significant.

To determine whether the difference between the rates of occurrence 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 H0)
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 rates does not equal the hypothesized difference. If you did not specify a hypothesized difference, Minitab tests whether there is no difference between the rates (Hypothesized difference = 0). 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 H0)
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 population rates are different. 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 2-Sample Poisson Rate.

When the hypothesized difference equals 0, Minitab tests the null hypothesis with an exact procedure. The p-value for the exact test is the result of this exact procedure. The other p-value is based on the normal approximation, and may be inaccurate when the total number of occurrences is low.

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