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An article in Air and Waste ["Update on Ozone Trends in California's South Coast Air Basin" (Vol. 43,1993 ) \(]\) studied the ozone levels on the South Coast air basin of California for the years \(1976-1991\). The author believes that the number of days that the ozone level exceeds 0.20 parts per million depends on the seasonal meteorological index (the seasonal average 850 millibar temperature). The data follow: $$\begin{array}{rrrrrr}\hline \text { Year } & \text { Days } & \text { Index } & \text { Year } & \text { Days } & \text { Index } \\\\\hline 1976 & 91 & 16.7 & 1984 & 81 & 18.0 \\\1977 & 105 & 17.1 & 1985 & 65 & 17.2 \\\1978 & 106 & 18.2 & 1986 & 61 & 16.9 \\\1979 & 108 & 18.1 & 1987 & 48 & 17.1 \\\1980 & 88 & 17.2 & 1988 & 61 & 18.2 \\\1981 & 91 & 18.2 & 1989 & 43 & 17.3 \\\1982 & 58 & 16.0 & 1990 & 33 & 17.5 \\\1983 & 82 & 17.2 & 1991 & 36 & 16.6 \\\\\hline\end{array}$$ (a) Construct a scatter diagram of the data. (b) Fit a simple linear regression model to the data. Test for significance of regression. (c) Find a \(95 \% \mathrm{CI}\) on the slope \(\beta_{1}\). (d) Analyze the residuals and comment on model adequacy.

Step by step solution

01

Construct the Scatter Diagram

Plot the data on a graph with the 'Index' as the X-axis and 'Days' as the Y-axis. Each point on the graph represents a year's corresponding index and number of days with high ozone levels. This helps in visualizing the potential linear relationship between the index and the number of ozone days.

02

Fit the Linear Regression Model

Calculate the line of best fit using the method of least squares. The regression line has the equation \( y = \beta_0 + \beta_1x \), where \( y \) is the number of days and \( x \) is the index. Here, \( \beta_0 \) is the y-intercept and \( \beta_1 \) is the slope.

03

Test Significance of Regression

Perform a hypothesis test on \( \beta_1 \) where the null hypothesis \( H_0: \beta_1 = 0 \) (no relationship) is tested against the alternative hypothesis \( H_a: \beta_1 eq 0 \) (relationship exists). Calculate the p-value from the t-distribution to decide if the null hypothesis can be rejected.

04

Calculate the 95% Confidence Interval for \( \beta_1 \)

Use the formula \( \beta_1 \pm t_{(n-2, 0.025)} \times SE(\beta_1) \) to calculate the confidence interval, where \( SE(\beta_1) \) is the standard error of the slope and \( t_{(n-2, 0.025)} \) is the critical value from the t-distribution.

05

Analyze the Residuals

Plot the residuals (differences between observed and predicted values) and check for any patterns. Analyze residual plots, such as residuals vs. fitted values and normal probability plot, to assess if the linear model fits well. Patterns may indicate model inadequacy.

06

Comment on Model Adequacy

Based on the residual analysis, comment on whether the linear regression model is a suitable fit for the data. Look for randomness in the residual plots which suggests that the model assumptions are satisfied.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scatter Plot

A scatter plot is a type of data visualization that plots data points on a two-dimensional graph. Here, each point represents a specific observation. To create a scatter plot of the given dataset, plot the 'Index' on the X-axis and 'Days' on the Y-axis.

This visualization helps you see whether there's a potential linear relationship between the seasonal meteorological index and the number of days the ozone level exceeds the set threshold.

• If the points tend to form a pattern stretching in a specific direction, it indicates some form of relationship.
• A scattered, random pattern would suggest no relationship.

In this case, the scatter plot might show a downward trend, hinting at a possible inverse relationship; as the index increases, the number of high ozone days declines.

Hypothesis Testing

In linear regression, hypothesis testing is used to determine if there's a meaningful relationship between two variables. The focus is on testing the slope of the regression line, typically denoted as \( \beta_1 \).

For hypothesis testing in this scenario:

• The null hypothesis \( H_0: \beta_1 = 0 \) suggests there's no relationship between the index and ozone days.
• The alternative hypothesis \( H_a: \beta_1 eq 0 \) implies a significant relationship exists.

By performing a t-test, you calculate a p-value that indicates the likelihood of observing the data if the null hypothesis were true.

A small p-value (usually less than 0.05) leads to rejecting the null hypothesis, supporting the idea that the seasonal index affects ozone days.

Confidence Interval

A confidence interval gives an estimated range of values which is likely to include an unknown population parameter. For the slope \( \beta_1 \) in regression analysis, it helps determine the precision of the slope estimate.

The formula for a 95% confidence interval for the slope is:\[\beta_1 \pm t_{(n-2, 0.025)} \times SE(\beta_1)\]
Where:

• \( \beta_1 \) is the estimated slope.
• \( t_{(n-2, 0.025)} \) is the critical value from the t-distribution.
• \( SE(\beta_1) \) is the standard error of the slope estimate.

If the confidence interval includes zero, it suggests that \( \beta_1 \) may not be significantly different from zero, indicating a potential lack of relationship.

Conversely, if zero is not within the interval, it suggests that the relationship is statistically significant, reinforcing the findings from hypothesis testing.

Residual Analysis

Residual analysis involves evaluating the differences between observed values and predicted values (residuals) to assess the quality of the regression model.

By plotting the residuals, you can visually check whether the model is appropriate. Some key things to look for include:

• A random scatter of residuals around the horizontal axis suggests a good fit.
• Patterns or systematic structures in the residual plot indicate model inadequacy.

Additionally, check the normal probability plot of residuals to ensure they are normally distributed.

If the residual analysis shows randomness and normality, it suggests that the linear assumptions are not violated and the model is a good fit for the data.