Problem 36 - An Introduction to Mathematical Statistics and Its Applications

Question

The data in the table examine the relationship between stock market changes (1) during the first few days in January and (2) over the course of the entire year. Included are the years from 1950 through \(1986 .\) (a) Use Theorem 14.6.1 to test the randomness of the January changes (relative to the number of runs up and down). Let \(\alpha=0.05\). (b) Use Theorem 14.6.1 to test the randomness of the annual changes. Let \(\alpha=0.05\). | | % Change for | | | :--- | :---: | :--- | | | First 5 Days in | % Change | | Year | Jan., x | for Year, y | | 1950 | 2.0 | 21.8 | | :--- | ---: | ---: | | 1951 | 2.3 | 16.5 | | 1952 | 0.6 | 11.8 | | 1953 | -0.9 | -6.6 | | 1954 | 0.5 | 45.0 | | 1955 | -1.8 | 26.4 | | 1956 | -2.1 | 2.6 | | 1957 | -0.9 | -14.3 | | 1958 | 2.5 | 38.1 | | 1959 | 0.3 | 8.5 | | 1960 | -0.7 | -3.0 | | 1961 | 1.2 | 23.1 | | 1962 | -3.4 | -11.8 | | 1963 | 2.6 | 18.9 | | 1964 | 1.3 | 13.0 | | 1965 | 0.7 | 9.1 | | 1966 | 0.8 | -13.1 | | 1967 | 3.1 | 20.1 | | 1968 | 0.2 | 7.7 | | 1969 | -2.9 | -11.4 | | ---: | ---: | ---: | | 1970 | 0.7 | 0.1 | | 1971 | 0.0 | 10.8 | | 1972 | 1.4 | 15.6 | | 1973 | 1.5 | -17.4 | | 1974 | -1.5 | -29.7 | | 1975 | 2.2 | 31.5 | | 1976 | 4.9 | 19.1 | | 1977 | -2.3 | -11.5 | | 1978 | -4.6 | 1.1 | | 1979 | 2.8 | 12.3 | | 1980 | 0.9 | 25.8 | | 1981 | -2.0 | -9.7 | | 1982 | -2.4 | 14.8 | | 1983 | 3.2 | 17.3 | | 1984 | 2.4 | 1.4 | | 1985 | -1.9 | 26.3 | | 1986 | -1.6 | 14.6 |

Step-by-Step Solution

Answer

The short answer will depend on the calculations performed in Steps 2-5. This includes calculating the actual and expected number of runs, the variance of the number of runs, finding the z-score, and finally the p-value. If the p-value is less than 0.05, the null hypothesis that the sequence is random is rejected.

Step 1: Define a Run

According to Theorem 14.6.1, a 'run' is defined as a sequence of similar items that are followed and preceded by a different item or no item at all. Here, one might define a run as a sequence of either increases or decreases in the percentage changes.

Step 2: Calculate the Number of Runs

Start by counting the number of runs for both the January changes and the annual changes. One can do this by going year by year and marking a new run each time the consecutive sequence of increases or decreases is broken.

Step 3: Calculate the Expected Number of Runs

We calculate the expected number of runs using the formula given by Theorem 14.6.1. The formula is \(E[R] = 2n_1n_2/n + 1\), where \(n_1\) is the number of positive percentage changes, \(n_2\) is the number of negative percentage changes, and \(n\) is the total number of years.

Step 4: Calculate the Variance of the Number of Runs

The variance of the number of runs can be calculated using the formula given by Theorem 14.6.1 : \(Var[R] = [2n_1n_2(2n_1n_2 - n)]/[n^2(n - 1)]\)

Step 5: Apply Theorem 14.6.1

The theorem tells us that if the population is randomly ordered, then approximately \(R - E[R] / sqrt(Var[R])\) will follow a standard normal distribution. This means we can calculate a z-score and use a standard normal table to find a p-value associated with this z-score. If the p-value is less than the significance level of 0.05, then we can conclude that there is enough evidence to reject the null hypothesis that the sequence is random.

Key Concepts

Theorem 14.6.1

To understand the concept of randomness in sequences, like stock market changes, we use a statistical tool called Theorem 14.6.1. This theorem plays a crucial role in a run test for randomness, which is used to analyze sequences in data to determine if a pattern is random or not. A 'run', in this context, is a consecutive series of either increases or decreases in data values, like price changes.

The theorem provides us with the mathematics to calculate the expected number of runs in a sequence given the number of positive and negative changes. Once determined, it gives us a way to compute the variance of the number of runs. These calculations help in understanding whether the runs in our data are random or exhibit an unusual pattern, possibly indicating a trend or cyclical behavior.

To execute Theorem 14.6.1 in a practical scenario, follow this process: count the actual number of runs in your data, calculate the expected number of runs and the variance using the formulas provided by the theorem. The next step involves converting the difference between actual and expected runs to a z-score, which can be compared to a standard normal distribution to infer randomness. This process assists in translating abstract statistical concepts into actionable insights in various fields, including finance and economics. Remember that in real-world applications, this tool allows us to test the hypothesis that sequences with too few or too many runs are not just due to randomness, and that perhaps an underlying cause must be investigated.

Hypothesis Testing

Hypothesis testing is a fundamental concept in statistics used to make inferences about populations based on sample data. In essence, hypothesis testing decides whether there is enough evidence in a sample to support a certain belief about the population from which the sample was drawn.

The process begins with the establishment of two opposing hypotheses: the null hypothesis (\(H_0\), which usually represents the status quo or a statement of 'no effect'), and the alternative hypothesis (\(H_a\) or (\(H_1\)), which represents a new claim aiming to challenge the status quo.

To test these hypotheses, we calculate a test statistic from our sample data—like the z-score mentioned in Theorem 14.6.1—which we then compare to a known distribution, such as the standard normal distribution in many cases. This comparison yields a p-value, which helps determine whether the evidence is strong enough to reject the null hypothesis for the alternative.

This method is not just for academic exercises; it is widely used in business, science, and policymaking to make informed decisions based on data. Whether it's determining if a new drug is effective or if a change in a business process has improved efficiency, hypothesis testing provides a structured way to draw conclusions from data.

Standard Normal Distribution

Picture a bell curve—this is what we call the standard normal distribution. It's a critical concept in statistics, representing a distribution that has been standardized so that it has a mean of zero and a standard deviation of one. Put simply, it serves as a reference to determine how unusual or typical a piece of data is within a dataset.

This distribution is the benchmark for the z-score calculation, which tells you how many standard deviations an element is from the mean. Since the areas under the curve correspond to probabilities, we can use z-scores to calculate the probability of observing a value at least as extreme as the test statistic, assuming the null hypothesis is true—this is the p-value.

In practice, the standard normal distribution is the basis for many statistical methods, including hypothesis testing and confidence intervals. Understanding how to use it allows you to tap into a wealth of statistical tools for interpreting data and making predictions. For instance, in finance, it can be used to assess the risk of investments, while in quality control, it helps monitor production processes. And in our context of runs test for randomness, the standard normal distribution provides the framework for interpreting the z-score and deciding if the observed data follows a random pattern.