Question

Cavities in Fourth-Grade Students A school dentist wanted to test the claim, at \(\alpha=0.05,\) that the number of cavities in fourth-grade students is random. Forty students were checked, and the number of cavities each had is shown here. Test for randomness of the values above or below the median. $$ \begin{array}{lllllllllll} 0 & 4 & 6 & 0 & 6 & 2 & 5 & 3 & 1 & 5 & 1 \\ 2 & 2 & 1 & 3 & 7 & 3 & 6 & 0 & 2 & 6 & 0 \\ 2 & 3 & 1 & 5 & 2 & 1 & 3 & 0 & 2 & 3 & 7 \\ 3 & 1 & 5 & 1 & 1 & 2 & 2 & & & & \end{array} $$

Short answer

The sequence of cavities is not random at \(\alpha = 0.05\).

Step-by-step solution

Organize the Data

First, we need to sort the data to find the median. The data are: [0, 4, 6, 0, 6, 2, 5, 3, 1, 5, 1, 2, 2, 1, 3, 7, 3, 6, 0, 2, 6, 0, 2, 3, 1, 5, 2, 1, 3, 0, 2, 3, 7, 3, 1, 5, 1, 1, 2, 2]. Sort them in ascending order: [0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7].

Find the Median

With the sorted list, find the median of the data. Since there are 40 values, the median is the average of the 20th and 21st values. The 20th value is 3, and the 21st value is also 3. Therefore, the median is \[\text{Median} = \frac{3 + 3}{2} = 3.\]

Count Above and Below Median

Count how many values are above and below the median. Values above 3: [4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7]. Counting these, there are 13 values above the median. Values below 3: [0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2]. Counting these, there are 18 values below the median.

Conduct the Runs Test

A runs test determines the randomness of a sequence by examining the number of runs of above-median and below-median values. A run is a sequence of similar elements. Convert the original data sequence into 'above' and 'below' median: B, A, A, B, A, B, A, B, B, A, B, B, B, B, B, A, B, A, B, B, A, A, B, B, B, A, B, B, B, B, B, B, A, A, A, B, A, A. Count the runs: there are **24 runs.**

Determine Expected Runs and Standard Deviation

Use the formula to calculate expected number of runs \( E(R) \) and standard deviation of runs \( \sigma(R) \) for a sequence:\[E(R) = \frac{2n_1n_2}{n_1 + n_2} + 1\]\[\sigma(R) = \sqrt{\frac{2n_1n_2(2n_1n_2-n_1-n_2)}{(n_1+n_2)^2(n_1+n_2-1)}}\]Where \(n_1 = 18\), \(n_2 = 13\). Calculate:\[E(R) = \frac{2(18)(13)}{31} + 1 \approx 15.5\]\[\sigma(R) = \sqrt{\frac{2(18)(13)(2(18)(13)-18-13)}{(31)^2(31-1)}} \approx 3.94\]

Calculate Z-value and Decision

We calculate the Z-value to compare with the critical Z-value at \(\alpha = 0.05\) (1.96 for two-tailed test):\[Z = \frac{R - E(R)}{\sigma(R)} = \frac{24 - 15.5}{3.94} \approx 2.16\]Since \(Z = 2.16\) is greater than 1.96, we have sufficient evidence to reject the null hypothesis of randomness.

Key concepts

Randomness Testing

Randomness testing is an important statistical method to determine if a sequence of data points occurs in a random pattern. Random sequences do not show any predictable patterns or trends. In the context of statistical analysis, randomness is key to ensuring that data points do not follow a bias or trend, which can distort the outcome of analysis and experiments. For example, in our exercise, the school dentist wants to test whether the number of cavities in students is random, which might imply they occur independently across the students rather than being influenced by common factors. To test this, different techniques, such as runs tests, are employed. A key aspect of randomness testing involves setting up a null hypothesis that assumes randomness and using sample data to challenge this assumption.

Typical methods for testing randomness include:

• Visual Inspection: Observing a graph or plot of data to see any patterns.
• Statistical Tests: Applying formal statistical tests like runs tests to detect randomness or patterns.

These methods are crucial in ensuring the integrity and validity of statistical analyses.

Median Calculation

Finding the median is a fundamental process in statistics which allows us to determine the central tendency of a dataset. The median represents the middle value when the data is sorted in ascending order. Unlike the mean, the median is not affected by extreme values, making it a better measure of central tendency in skewed data. In this exercise, calculating the median of the number of cavities among students helps divvy up the data into parts "above median" and "below median" for further analyses.

The steps to calculate the median are simple:

• Sort the dataset in ascending order.
• Identify the middle value. If the number of observations is odd, it's the center value; if even, it's the average of the two central values.

For instance, with our 40 students’ cavity numbers, the middle values (20th and 21st) are both 3. Thus, the median is 3. Calculating the median accurately is critical, especially when preparing for randomness tests where the data is split based on this value.

Runs Test

A runs test is a statistical procedure used to analyze the sequence of observations to determine randomness. This specific type of test is crucial when you need to ensure that the data sequence is not ordered in a specific way. The concept of a "run" is a sequence of similar items or categories.

In our step-by-step solution, once we determine data above and below the median, these observations can be coded as 'A' for above, and 'B' for below. By counting the number of runs within this coded sequence, we can test how likely the sequence was random. In the example given, there are 24 runs, which exceed the expected number of random runs.

Running the test follows these steps:

• Identify and label data points as belonging to one of two categories.
• Count the number of contiguous sequences, or runs, for each category.
• Compare the observed number of runs to the expected number using statistical formulas.

The runs test provides a practical way to assess whether a random process like student cavities across a sample is involved.

Hypothesis Testing

Hypothesis testing is a crucial part of statistics used to determine the presence of effects or patterns in your data. It involves making an initial assumption, called the null hypothesis, which states that there is no effect or relationship, such as 'the number of cavities is random'. In contrast, the alternative hypothesis states that there is an effect, or a relationship is present.

The processes involved typically include:

• Formulating null and alternative hypotheses.
• Choosing a significance level (like \(\alpha = 0.05\)).
• Calculating a test statistic to see how your results compare to what would be expected under the null hypothesis.
• Using a critical value to decide whether to reject the null hypothesis.

In our exercise, the test statistic Z-value was 2.16, which at a significance level of 0.05, led to rejecting the null hypothesis of randomness since it was greater than the critical value of 1.96. Hypothesis testing allows researchers to make data-driven decisions and inferences about the broader population based on sample data.