Question

{ 7.29 Random } & \text { Digits in } & \text { Students' } & \text { Random }\end{array}\( Numbers? How well can people generate random numbers? A sample of students were asked to write down a "random" four-digit number. Responses from 150 students are stored in the file Digits. The data file has separate variables (RND1, RND2, \)R N D 3,\( and \)R N D 4\( ) containing the digits in each of the four positions. (a) If the numbers are randomly generated, we would expect the last digit to have an equal chance of being any of the 10 digits. Test \)H_{0}\( : \)p_{0}=p_{1}=p_{2}=\cdots=p_{9}=0.10\( using technology and the data in \)R N D 4\(. (b) Since students were asked to produce four-digit numbers, there are only nine possibilities for the first digit (zero is excluded). Use technology to test whether there is evidence in the values of \)R N D 1$ that the first digits are not being chosen by students at random.

Short answer

The randomness of a four-digit number generated by students is validated by testing the hypothesis if all possible digits in each position are equally likely (uniform distribution). The Chi-Square goodness-of-fit test is used to test the null hypothesis. For RND4 (last digit), we test \(p_{0}=p_{1}=p_{2}=...=p_{9}=0.10\). For RND1 (first digit), we test if \(p_{1}=p_{2}=...=p_{9}\), given that the first digit cannot be zero. A p-value less than our selected significance level (e.g., 0.05) is a reason to reject the null hypothesis, implying the digits aren't randomly generated.

Step-by-step solution

Data understanding and hypothesis testing for RND4

First, let's understand the data, especially the 'RND4' variable which represents the fourth or the last digit in the four-digit randomly generated number from student inputs. We will conduct a hypothesis test under the null hypothesis \(H_{0}\): \(p_{0}=p_{1}=p_{2}=...=p_{9}=0.10\), i.e., all digits from 0 to 9 are equally likely. It is necessary to obtain frequency count for each digit in the RND4 column and calculate the observed frequencies. Use an appropriate test, like the Chi-Square goodness-of-fit test, to test for the equality of proportions.

Analyzing the results for RND4

Analyze the result from the Chi-Square goodness-of-fit test. If the obtained p-value is less than the significance level (e.g., 0.05), we reject the null hypothesis and conclude that the last digit is not uniformly distributed. This means it is not randomly generated as expected.

Data understanding and hypothesis testing for RND1

Following similar steps as before, let's understand the 'RND1' variable which represents the first digit in the four-digit randomly generated number. Since the first digit can't be zero in a four-digit number, the possible values range from 1 to 9. We will test the null hypothesis \(H_{0}\): \(p_{1}=p_{2}=...=p_{9}\), i.e., all digits from 1 to 9 are equally likely.

Analyzing the results for RND1

Similar to Step 2, analyze the result from the Chi-Square goodness-of-fit test. If the obtained p-value is less than the significance level (e.g., 0.05), we reject the null hypothesis and conclude that the first digit is not uniformly distributed; hence, it is not randomly generated as expected.

Key concepts

Random Number Generation

Have you ever wondered how computationally it's possible to create a number that doesn't follow a predictable pattern? Random number generation is a process used in various fields such as cryptography, simulations, and statistical sampling to produce a sequence of numbers that lacks any predictable order. These numbers are essential for many statistical methods because they replicate the concept of 'chance' or 'randomness' in a controlled environment.

In educational settings, understanding how to generate random numbers can illuminate the study of probability and statistics. When students attempt to create random numbers themselves, as in our exercise, it highlights the psychological tendencies and biases humans have, as opposed to the unbiased randomness a computer algorithm can produce. While students might believe they can think of numbers 'at random,' our innate biases often lead to patterns that wouldn't occur in truly random sequences, which is part of what we're testing in the provided exercise.

Chi-Square Goodness-of-Fit Test

The chi-square goodness-of-fit test is a non-parametric statistical hypothesis test used to determine whether a set of observed frequencies differs significantly from the expected frequencies. It is a useful tool to assess how well data fits a specific distribution or to evaluate if categorical variables are independent.

In the context of our exercise where student-generated numbers are being assessed, the chi-square test can help to analyze if the observed distribution of the last digit (RND4) or the first digit (RND1) matches the expected uniform distribution - where each digit is theoretically equally likely to appear. This test calculates a chi-square statistic that measures the divergence of the observed frequencies from the expected ones, and with the help of a p-value, it guides us whether to accept or reject our null hypothesis.

Null Hypothesis

The core of hypothesis testing starts with setting a default claim or a hypothesis, known as the null hypothesis, symbolized by H₀(H-naught). This hypothesis posits no effect or no difference, meaning you assume there is no significant change or peculiar pattern unless proven otherwise. In our example, the null hypothesis for the last digit (RND4) assumes that every digit from 0 to 9 has an equal chance of 10% to be chosen (H₀: p₀=p₁=...=p₉=0.10).

For hypothesis testing, scientists and statisticians set up a counter hypothesis as well, known as the alternative hypothesis, which directly contradicts the null. After performing the appropriate test, if evidence is insufficient to support the alternative hypothesis, the null stands; however, if significant evidence contradicts the null, we reject it in favor of the alternative hypothesis.

Observed Frequencies

In the realm of statistics, observed frequencies refer to the count of occurrences that actually happen in the collected data. It's the real-life tally and not what we might assume or predict to happen merely based on probability theory. In practice, after we collect data like students' 'randomly' chosen digits, we count how many times each digit appears - these are our observed frequencies, which we then compare to our expected frequencies in hypothesis testing.

In our exercise, the observed frequencies are crucial for applying the chi-square goodness-of-fit test. We use them to gauge whether there's a discrepancy between our observed data (how many times each digit was actually chosen) and what we would expect if digits were chosen with uniform probability. If the frequencies show a pattern or deviation from the uniform distribution, it can indicate that students' choices aren't as random as one might assume.