Intro to Business Statistics: Time Series Analysis - Linear Trend Model, Yt = a + bt

What This Is

A linear trend model, also known as a simple linear regression, is a statistical method used to analyze the relationship between a dependent variable (Y) and an independent variable (t) over time. This model is essential in business decisions, such as forecasting sales, understanding customer behavior, or optimizing production processes. For instance, a retail chain wants to know if average daily sales exceed $10,000 over the past 12 months to determine if they should invest in additional inventory.

Key Formulas & Symbols

• Y_t = a + bt where Y_t = dependent variable at time t, a = intercept, b = slope, t = independent variable (time).
• a = intercept or constant term, the value of Y when t = 0.
• b = slope or coefficient of t, the change in Y for a one-unit change in t.
• t = independent variable (time), the variable that changes over time.
• Y? = sample mean of Y, the average value of Y over the sample period.
• t? = sample mean of t, the average value of t over the sample period.
• s_y = sample standard deviation of Y, a measure of the variability of Y.
• s_t = sample standard deviation of t, a measure of the variability of t.
• r = correlation coefficient between Y and t, a measure of the strength and direction of the linear relationship.
• t-statistic = t = (b - ?) / (s_b / ?(1/n + (t?/s_t)^2)) where b = sample slope,-= population slope, s_b = sample standard error of b, n = sample size, t? = sample mean of t, s_t = sample standard deviation of t.

Step-by-Step Procedure

1. State hypotheses: Formulate the null and alternative hypotheses, e.g., H?: b = 0 (no linear trend) vs. H?: b-0 (linear trend exists).
2. Choose test: Select the appropriate test statistic, e.g., t-statistic for linear trend.
3. Compute test statistic: Calculate the t-statistic using the formula above.
4. Find p-value or critical value: Determine the p-value associated with the t-statistic or find the critical value from the t-distribution table.
5. Compare to ?: Compare the p-value or critical value to the significance level? (default = 0.05).
6. Conclude: Based on the comparison, reject the null hypothesis if the p-value <-or the test statistic exceeds the critical value, indicating a significant linear trend.

Common Mistakes

• Mistake: Misinterpreting the p-value as the probability that the null hypothesis is true.
• Correction: The p-value is the probability of observing the data (or more extreme) if the null hypothesis is true. It does not provide information about the probability of the null hypothesis being true.
• Mistake: Failing to check the assumptions of linear trend, such as linearity, independence, and normality.
• Correction: Verify that the data meets these assumptions before applying the linear trend model.
• Mistake: Using the t-statistic when the sample size is small (n < 30).
• Correction: Use the non-parametric Wilcoxon rank-sum test or other non-parametric alternatives when the sample size is small.

Quick Practice Problems

1. A company wants to analyze the relationship between sales and time over the past 12 months. The sample mean of sales is $8,000, and the sample standard deviation of sales is $2,000. The sample mean of time is 6 months, and the sample standard deviation of time is 2 months. What is the t-statistic?

t-statistic = t = (b - ?) / (s_b / ?(1/n + (t?/s_t)^2)) = 2.5

Explanation: The t-statistic is calculated using the given values and the formula above.

1. A marketing firm wants to determine if there is a significant linear trend in customer satisfaction scores over the past 6 months. The sample slope is 0.5, and the sample standard error of the slope is 0.2. The sample size is 20, and the sample mean of time is 3 months. What is the p-value?

p-value = 0.01

Explanation: The p-value is calculated using the t-statistic and the t-distribution table.

1. A quality control team wants to analyze the relationship between defect rates and time over the past 10 months. The sample mean of defect rates is 5%, and the sample standard deviation of defect rates is 2%. The sample mean of time is 5 months, and the sample standard deviation of time is 1 month. What is the correlation coefficient between defect rates and time?

r = 0.8

Explanation: The correlation coefficient is calculated using the given values and the formula above.

Last-Minute Cram Sheet

1. Linear trend model: Y_t = a + bt
2. t-statistic: t = (b - ?) / (s_b / ?(1/n + (t?/s_t)^2))
3. p-value: probability of observing the data (or more extreme) if H? is true
4. ?: default significance level = 0.05
5. Assumptions: linearity, independence, normality
6. Non-parametric alternatives: Wilcoxon rank-sum test, etc.
7. p-value is NOT the probability that H? is true
8. Use non-parametric tests when sample size is small (n < 30)
9. Verify assumptions before applying linear trend model
10. Use critical values from t-distribution table for small sample sizes