Understanding Significance Tests for Sample Mean

The question asks us to identify the correct significance test for a sample of mean annual rainfall data. We are given specific details about the sample and the population it was drawn from. To choose the correct test, we need to consider several factors related to the data.

Analyzing the Given Information

Let's list the key information provided:

• Sample size (n): 20 places
• Population mean annual rainfall ($\mu$): 320 cm (This is a known population mean)
• Sample mean annual rainfall ($\bar{x}$): 250 cm
• Sample standard deviation (s): 150 cm
• Population distribution: Normally distributed universe

We need to test if the sample mean of 250 cm is significantly different from the population mean of 320 cm. This is a hypothesis testing problem involving a single sample mean compared to a known population mean.

Choosing the Appropriate Significance Test

The choice of the appropriate significance test for comparing a sample mean to a population mean primarily depends on:

1. The sample size (n)
2. Whether the population standard deviation ($\sigma$) is known or unknown
3. The distribution of the population

Let's examine the conditions from the question against these criteria:

• Sample Size: The sample size is n = 20. This is considered a small sample size (typically n < 30).
• Population Standard Deviation: The population standard deviation ($\sigma$) is not given. We are given the sample standard deviation (s = 150 cm). So, $\sigma$ is unknown.
• Population Distribution: The question explicitly states that the sample is from a normally distributed universe.

Now let's consider the common tests for means:

• Z-test: The Z-test for a single mean is typically used when:
  • The population standard deviation ($\sigma$) is known, OR
  • The sample size is large (n ≥ 30), regardless of whether $\sigma$ is known (due to the Central Limit Theorem, which suggests the sample mean distribution is approximately normal).
  In this problem, the sample size is small (n=20) and the population standard deviation ($\sigma$) is unknown. Therefore, the Z-test is not the most appropriate test.
• t-test: The t-test for a single mean is typically used when:
  • The population standard deviation ($\sigma$) is unknown, AND
  • The sample size is small (n < 30), AND
  • The population is approximately normally distributed.
  In this problem, all three conditions are met: $\sigma$ is unknown, n=20 (small sample), and the population is normally distributed. This points towards the t-test.
• Chi-squared test ($\chi^2$ test): The Chi-squared test is used for categorical data, such as testing for independence between two categorical variables or testing if sample data fits a specific distribution (goodness-of-fit). It is not used for comparing means of continuous data like rainfall.
• F-test: The F-test is used to compare variances of two or more populations or samples. It is also used in ANOVA (Analysis of Variance) to compare means across multiple groups, but it focuses on the variances within and between groups. It is not used for comparing a single sample mean to a known population mean.

Based on the analysis of the sample characteristics (small sample, unknown population standard deviation) and the population distribution (normal), the t-test is the correct significance test for this scenario.

Conclusion

Given the sample size is 20 (small), the population standard deviation is unknown, and the population is normally distributed, the appropriate statistical test is the t-test for a single mean.

Revision Table: Significance Tests

This table summarizes the conditions for using common significance tests for means and variances.

Additional Information on Statistical Tests

Let's delve a bit deeper into the tests mentioned in the options.

t-test Explained

The t-test, specifically Student's t-test, is used when you need to make inferences about a population mean but do not know the population standard deviation ($\sigma$) and are working with a small sample size. It relies on the t-distribution, which is similar in shape to the normal distribution but has heavier tails, accounting for the increased uncertainty due to estimating the population standard deviation from the sample.

The formula for the t-statistic when comparing a sample mean ($\bar{x}$) to a known population mean ($\mu$) is:

\( t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \)

Where:

• $\bar{x}$ is the sample mean
• $\mu$ is the population mean
• s is the sample standard deviation
• n is the sample size

The degrees of freedom for this test are \(df = n - 1\).

Z-test Explained

The Z-test is used when the population standard deviation is known, or when the sample size is large enough (usually n ≥ 30) that the sample standard deviation is a reliable estimate of the population standard deviation and the sampling distribution of the mean can be approximated by a normal distribution (Central Limit Theorem). The Z-statistic is calculated similarly to the t-statistic, but uses $\sigma$ instead of s.

\( Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \)

Chi-squared Test Explained

The Chi-squared ($\chi^2$) test is a non-parametric test used to analyze categorical data. It assesses how likely it is that any observed difference between sets of categorical data arose by chance. Common uses include testing if observed frequencies differ significantly from expected frequencies (goodness-of-fit) or testing if two categorical variables are independent (test of independence).

F-test Explained

The F-test is based on the F-distribution and is primarily used to compare the variances of two populations. It is also the basis for Analysis of Variance (ANOVA), a technique used to compare means across three or more groups by partitioning the total variability in the data into components attributed to different sources.