Interpret all statistics for 1-Sample Poisson Rate

Find definitions and interpretation guidance for every statistic that is provided with the 1-sample Poisson rate test.

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 a box of towels. A towel can have more than one defect, such as 1 tear and 2 pulls (3 defects). Each box contains 10 towels. The inspectors sample 50 total boxes, and find a total of 122 defects.
  • The total occurrences is 122 because the inspectors find 122 defects.
  • The sample size (N) is 50 because the inspectors sample 50 boxes.
  • To determine the number of defects per towel, inspectors use an observation length of 10 because each box contains 10 towels. To determine the number of defects per box, inspectors use an observation length of 1.
  • The sample rate is (Total occurrences / N) / (observation length) = (122/50) / 10 = 0.244. So, on average, each towel has 0.244 defects.

Null hypothesis and alternative hypothesis

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 hypothesized rate.

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 a box of towels. A towel can have more than one defect, such as 1 tear and 2 pulls (3 defects). Each box contains 10 towels. The inspectors sample 50 total boxes, and find a total of 122 defects.
  • The total occurrences is 122 because the inspectors find 122 defects.
  • The sample size (N) is 50 because the inspectors sample 50 boxes.
  • To determine the number of defects per towel, inspectors use an observation length of 10 because each box contains 10 towels. To determine the number of defects per box, inspectors use an observation length of 1.
  • The sample rate is (Total occurrences / N) / (observation length) = (122/50) / 10 = 0.244. So, on average, each towel has 0.244 defects.

N

The sample size (N) is the number of times that you count occurrences 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 a box of towels. A towel can have more than one defect, such as 1 tear and 2 pulls (3 defects). Each box contains 10 towels. The inspectors sample 50 total boxes, and find a total of 122 defects.
  • The total occurrences is 122 because the inspectors find 122 defects.
  • The sample size (N) is 50 because the inspectors sample 50 boxes.
  • To determine the number of defects per towel, inspectors use an observation length of 10 because each box contains 10 towels. To determine the number of defects per box, inspectors use an observation length of 1.
  • The sample rate is (Total occurrences / N) / (observation length) = (122/50) / 10 = 0.244. So, on average, each towel has 0.244 defects.

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.

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 a box of towels. A towel can have more than one defect, such as 1 tear and 2 pulls (3 defects). Each box contains 10 towels. The inspectors sample 50 total boxes, and find a total of 122 defects.
  • The total occurrences is 122 because the inspectors find 122 defects.
  • The sample size (N) is 50 because the inspectors sample 50 boxes.
  • To determine the number of defects per towel, inspectors use an observation length of 10 because each box contains 10 towels. To determine the number of defects per box, inspectors use an observation length of 1.
  • The sample rate is (Total occurrences / N) / (observation length) = (122/50) / 10 = 0.244. So, on average, each towel has 0.244 defects.

Interpretation

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.

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.

Confidence interval (CI) and bounds

The confidence interval provides a range of likely values for the population rate. 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 rate. The percentage of these confidence intervals or bounds that contain the rate 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 rate.

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

Descriptive Statistics Total N Occurrences Sample Rate 95% CI for λ 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.

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.

You must choose Normal approximation as the method for Minitab to calculate the Z–value.

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 the 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 it is not, 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 population rate is statistically different from the hypothesized rate.

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 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 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 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 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.
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