Inferences for Slope: Videos & Practice Problems

Hypothesis testing and confidence intervals for beta, the slope of the population's regression line, assess linear correlation between variables. Using TI-84 functions like LinRegTTest and LinRegTInt, students can test null hypotheses (β = 0) against alternatives (β ≠ 0) at significance level $\alpha$. A confidence interval excluding zero indicates significant linear correlation. These inferential statistics methods help determine if the regression slope differs from zero, confirming relationships between independent and dependent variables in regression analysis.

Hypothesis Test for the Slope of a Regression Line

Understanding the relationship between two variables often involves analyzing the slope of the population's regression line, denoted as β. The slope β indicates the strength and direction of a linear correlation between variables. To determine if an observed linear trend in sample data is statistically significant and can be generalized to the population, a hypothesis test for β is conducted.

In this context, the null hypothesis (H₀) always states that there is no linear correlation between the variables, meaning the population slope β = 0. The alternative hypothesis (H_a) depends on the research question: it can be two-sided (β ≠ 0) if testing for any linear relationship, or one-sided (β > 0 or β < 0) if testing for a positive or negative correlation specifically.

For example, consider an economic study examining the relationship between advertising spending and product sales (measured in thousands of dollars). To test the claim that no linear relationship exists at a significance level of α = 0.05, the hypotheses are set as:

H₀: β = 0
H_a: β ≠ 0

Using statistical software or a graphing calculator such as the TI-84, the data for advertising spending and sales are inputted into separate lists. The LinRegTTest function performs the hypothesis test for the slope β. This test calculates a t-statistic and corresponding p-value to assess the evidence against the null hypothesis.

When the p-value is less than the significance level α, the null hypothesis is rejected, indicating sufficient evidence that the population slope β is not zero. This implies a statistically significant linear correlation between the variables. For instance, a p-value of approximately 3.4 × 10⁻⁶ is much smaller than 0.05, leading to rejection of H₀ and supporting the conclusion that advertising spending and product sales are linearly related.

This hypothesis testing approach for the regression slope is essential in inferential statistics, allowing researchers to extend findings from sample data to broader populations. It complements correlation coefficient tests and provides a rigorous method to confirm whether observed trends reflect true relationships or random chance.

Using the sample data below, run a hypothesis test on $\beta$ to see if there is evidence that there is a positive correlation between $x$ and $y$ with $\alpha =0.01$.

• A. Reject $H_0$ and conclude that there is a positive correlation between $x$ and $y$ and that $\beta >0$.
• B. Fail to reject $H_0$ since there is enough evidence to suggest $\beta>0$, but not enough evidence to suggest positive linear correlation between $x$ and $y$.
• C. Fail to reject $H_0$ since there is not enough evidence to suggest $\beta >0$ and not enough evidence to suggest positive linear correlation between $x$ and $y$.
• D. Reject $H_0$ since there is not enough evidence to suggest $\beta>0$ and not enough evidence to suggest positive linear correlation between $x$ and $y$.

Answer: (C)

Hypothesis Test for the Slope of a Regression Line Example 1

When investigating the relationship between website loading speed (measured in seconds) and the number of visits per minute, it is essential to determine if a linear correlation exists. This involves performing a hypothesis test on the slope β of the population regression line. The null hypothesis (H₀) states that there is no linear correlation, meaning β = 0. The alternative hypothesis (H_a) posits that β ≠ 0, indicating a nonzero linear correlation between loading speed and visits.

To test these hypotheses, data on website speed and visits are input into statistical software or a graphing calculator, such as the TI-84. The loading speed data is entered into list 1 (L1) and the visits data into list 2 (L2). Using the WinRegTTest function, a test for the slope of the regression line is conducted with a significance level (α) of 0.1. This test calculates a p-value, which quantifies the probability of observing the data assuming the null hypothesis is true.

In this scenario, the calculated p-value is approximately 0.003, which is less than the significance level of 0.1. Since p < α, the null hypothesis is rejected, providing sufficient evidence to conclude that the slope β is not zero. This means there is a statistically significant linear correlation between website loading speed and the number of visits per minute.

Understanding this process highlights the importance of hypothesis testing in regression analysis, where the slope β represents the rate of change in the dependent variable (visits) with respect to the independent variable (loading speed). Rejecting the null hypothesis confirms that changes in loading speed are associated with changes in website visits, which can inform web design decisions aimed at optimizing user engagement.

Hypothesis Test for the Slope of a Regression Line Example 2

When investigating the relationship between sleep duration and reaction time, a hypothesis test can determine if increased sleep significantly decreases reaction time. This involves analyzing the slope β of the population regression line, which represents the linear correlation between the two variables. The null hypothesis (H₀) assumes no linear relationship, meaning β = 0. The alternative hypothesis (H_a) posits a negative linear correlation, where increased sleep leads to decreased reaction time, so β < 0.

To test these hypotheses, data on sleep hours and reaction times (in milliseconds) are collected and entered into statistical software or a graphing calculator, such as the TI-84. Sleep data is input into list L1 and reaction time data into list L2. Using the linear regression t-test function (LinRegTTest), the test is set up with the x-list as L1, y-list as L2, frequency as 1, and the alternative hypothesis specified as β < 0. This test evaluates whether the slope of the regression line is significantly less than zero at a chosen significance level, here α = 0.01.

The output provides a p-value, which quantifies the probability of observing the data assuming the null hypothesis is true. In this case, the p-value is approximately 1.75 × 10⁻⁶, which is much smaller than the significance level of 0.01. Since p < α, the null hypothesis is rejected, indicating strong evidence for a negative slope. This confirms a statistically significant negative linear correlation between sleep duration and reaction time, meaning that as sleep increases, reaction time tends to decrease.

This analysis highlights the importance of hypothesis testing in regression to understand relationships between variables. The slope β serves as a key parameter indicating the direction and strength of the linear association, and the p-value helps determine the statistical significance of this relationship. By rejecting the null hypothesis, we conclude that increased sleep is associated with faster reaction times, which can have practical implications for health and cognitive performance.

Hypothesis Test for the Slope of a Regression Line Example 3

A quality control expert investigates whether increasing the frequency of machine servicing in a plant improves each machine's productivity. Data collected includes the number of times a machine is serviced per week and its corresponding daily productivity. To analyze this relationship, a linear regression model is used, which predicts productivity (y) based on the number of services (x).

The key step involves testing the slope (β) of the population regression line to determine if there is a statistically significant positive correlation between servicing frequency and productivity. The null hypothesis (H₀) states that β = 0, meaning no linear relationship exists. The alternative hypothesis (H₁) posits that β > 0, indicating that more frequent servicing increases productivity.

Using statistical software or a graphing calculator, the data is input with servicing frequency as the independent variable (x) and productivity as the dependent variable (y). A linear regression t-test is performed at a significance level (α) of 0.05. The resulting p-value is approximately 0.09, which exceeds the alpha threshold. This means there is insufficient evidence to reject the null hypothesis, so the data does not support a positive linear correlation between servicing frequency and productivity.

From the regression output, the sample regression line is determined as:

\[ y = 22.43 + 1.23x \]

Here, 22.43 is the y-intercept, representing the estimated productivity when no servicing occurs, and 1.23 is the slope, indicating the estimated increase in productivity per additional service. However, since the hypothesis test did not find the slope to be significantly greater than zero, this increase is not statistically supported.

In conclusion, the company does not have sufficient evidence to claim that increasing the number of machine services will enhance productivity. This analysis highlights the importance of hypothesis testing in validating whether observed trends in sample data reflect true relationships in the population.

Confidence Interval for the Slope of a Regression Line

Constructing a confidence interval for β, the slope of the population regression line, provides a powerful method to assess whether two variables exhibit a linear relationship. This approach complements hypothesis testing by estimating a range of plausible values for β rather than testing a specific claim directly. For example, when analyzing the relationship between hours teachers spend on lesson plans and students' average test scores, a 95% confidence interval can be constructed to determine if the slope significantly differs from zero, indicating linear correlation.

To create this confidence interval, the confidence level (C) is set at 0.95, reflecting 95% certainty that the true slope lies within the calculated interval. Using statistical tools such as the TI-84 calculator, the LinRegTInt function efficiently computes this interval by inputting the explanatory variable data (hours) and response variable data (test scores). The calculator outputs a confidence interval, for instance, from approximately 0.93 to 1.5.

Interpreting this interval is crucial: since zero is not included within the range, it suggests that the slope β is significantly different from zero at the 95% confidence level. This exclusion of zero implies sufficient evidence to conclude a linear correlation between the variables. In regression analysis, a slope of zero would indicate no linear relationship, so a confidence interval that excludes zero confirms the presence of a statistically significant linear association.

This method highlights the importance of confidence intervals in inferential statistics, providing not only a test of significance but also an estimate of the effect size and its precision. By understanding and applying confidence intervals for the slope, one can make informed conclusions about the strength and direction of relationships in bivariate data, enhancing the interpretation of regression results.

Using the sample data below, create a confidence interval for $\beta$ to see if there is evidence that there is a positive correlation between $x$ and $y$ with $\alpha =0.01$.

• A. $(-0.75,1.37)$; there is not enough evidence to suggest that there is a positive correlation between $x$ and $y$ with $\alpha=0.01$.
• B. $\left(-0.75,1.37\right)$; there is enough evidence to suggest that there is a positive correlation between $x$ and $y$ with $\alpha=0.01$.
• C. $(0.75,1.37)$; there is not enough evidence to suggest that there is a positive correlation between $x$ and $y$ with $\alpha=0.01$.
• D. $\left(0.75,1.37\right)$; there is enough evidence to suggest that there is a positive correlation between $x$ and $y$ with $\alpha=0.01$.

Answer: (A)

Confidence Interval for the Slope of a Regression Line Example 4

When analyzing the relationship between the length of a television advertisement and the resulting sales in thousands of dollars, it is essential to understand how to construct and interpret a confidence interval for the slope (β) of the population regression line. The slope β represents the rate of change in sales with respect to ad length, indicating whether longer ads are associated with increased sales.

To estimate this relationship with a 90% confidence level (α = 0.1), a confidence interval for β is constructed using statistical software or a calculator such as the TI-84. The data for ad length and sales are input into two lists, typically labeled L1 and L2. Using the LinRegTInt function, which performs a linear regression t-interval, the confidence interval for β is calculated. This interval provides a range of plausible values for the true slope of the regression line in the population.

In this case, the confidence interval for β was found to be approximately from 0.1 to 0.3. Since the entire interval consists of positive values, it indicates that there is a positive linear correlation between ad length and sales. This means that as the length of the advertisement increases, sales tend to increase as well.

Because the confidence interval does not include zero and is strictly positive, we can be 90% confident that the true slope β is positive. This provides strong evidence to support the claim that longer advertisements lead to higher sales. Therefore, it is advisable for the advertising executive to produce longer ads to potentially boost product sales.

Understanding this concept involves recognizing that a confidence interval for the slope in linear regression quantifies the uncertainty around the estimated effect of the independent variable (ad length) on the dependent variable (sales). When the interval is entirely above zero, it confirms a statistically significant positive relationship, reinforcing the decision-making process based on data-driven insights.

Confidence Interval for the Slope of a Regression Line Example 5

When investigating the relationship between sleep duration and the number of typos made in emails, it is essential to understand how to interpret the slope of the population regression line. The slope, denoted as β, represents the rate of change in typos for each additional hour of sleep. To estimate this slope with a certain level of confidence, a confidence interval can be constructed. For example, with a significance level α = 0.1, the confidence level is calculated as 1 - α = 0.9, or 90%. This means we are 90% confident that the true slope lies within the interval calculated from the sample data.

Using statistical software or a graphing calculator, such as the TI-84, one can input the sleep data and typo counts into separate lists and perform a linear regression t-interval test (LoomRegTInt). This procedure outputs a confidence interval for the slope β. In this case, the interval was approximately from -2.2 to -1.6. Since the entire interval is negative, it indicates a negative linear relationship between hours of sleep and the number of typos.

This negative slope means that as sleep increases, the number of typos decreases, confirming the hypothesis that more sleep can reduce errors in typing. The confidence interval being entirely below zero provides strong evidence against the null hypothesis of no relationship or a positive relationship. Therefore, we can conclude with 90% confidence that increased sleep is associated with fewer typos, demonstrating a statistically significant negative correlation between these variables.

Confidence Interval for the Slope of a Regression Line Example 6

A film critic is investigating whether there is a linear relationship between movie length and the score a film receives. To analyze this, data from a random sample of movies, including their lengths and scores, is collected. The goal is to determine the regression line that best fits this sample data and to assess if the slope of the population regression line, denoted as β, indicates a significant association between movie length and score.

To begin, the confidence interval for the slope β is constructed with a significance level α = 0.01, corresponding to a 99% confidence level. Using statistical software or a graphing calculator such as the TI-84, the movie length data is entered as the independent variable list (L1) and the movie score data as the dependent variable list (L2). The linear regression t-interval function (LinRegTInt) is then used to calculate the confidence interval for the slope.

The resulting confidence interval for the slope ranges approximately from $-52$ to $30$. Alongside this, the regression equation for the sample data is found to be:

$$y\; =\; 97\; -\; 11x$$

Here, $y$ represents the predicted movie score, $x$ is the movie length, $97$ is the y-intercept, and $-11$ is the slope. The negative slope suggests that, in the sample, longer movies tend to have lower scores, but this relationship needs further evaluation.

To determine if the film critic should pursue further research, it is essential to interpret the confidence interval for the slope. Since the interval includes zero, it implies that the true slope β of the population regression line could be zero. A slope of zero indicates no linear correlation between movie length and score. Therefore, there is insufficient evidence to conclude a significant linear relationship exists between these variables.

In summary, the regression analysis shows a sample slope of approximately $-11$, but the 99% confidence interval for the population slope spans negative to positive values, including zero. This means the film critic should not proceed with further research based on the current data, as the evidence does not support a definitive linear association between movie length and film score.

Here’s what students ask on this topic:

What is the null hypothesis when testing the slope of a population regression line?

The null hypothesis in testing the slope $\beta$ of a population regression line is that there is no linear relationship between the two variables. This means the slope of the population regression line is zero, expressed as $H{0}_{}:\beta =0$. It implies that changes in the explanatory variable do not predict changes in the response variable. Rejecting this null hypothesis suggests a significant linear correlation exists between the variables.

How do you use a TI-84 calculator to perform a hypothesis test for the slope of a regression line?

To perform a hypothesis test for the slope $\beta$ using a TI-84, first enter your data into lists (e.g., $L1$ for x-values and $L2$ for y-values). Then, press STAT, navigate to the TESTS menu, and select LinRegTTest. Ensure the x-list and y-list are correctly set, frequency is 1, and choose the appropriate alternative hypothesis (e.g., $\beta \ne 0$ for two-tailed). Calculate to get the p-value. If the p-value is less than your significance level (commonly 0.05), reject the null hypothesis, indicating a significant linear relationship.

What does a confidence interval for the slope of a regression line tell us?

A confidence interval for the slope $\beta$ provides a range of plausible values for the true slope of the population regression line. For example, a 95% confidence interval means we are 95% confident that the true slope lies within this range. If the interval does not include zero, it suggests a significant linear relationship between the variables. Conversely, if zero is within the interval, we do not have enough evidence to conclude a linear correlation exists.

How do you construct a 95% confidence interval for the slope using a TI-84 calculator?

To construct a 95% confidence interval for the slope $\beta$ on a TI-84, enter your data into lists (e.g., $L1$ and $L2$). Press STAT, go to the TESTS menu, and select LinRegTInt. Confirm the x-list and y-list, set frequency to 1, and enter the confidence level as 0.95. Calculate to obtain the interval. If the interval excludes zero, it indicates a significant linear relationship between the variables.

What does it mean if the p-value from a hypothesis test for the slope is less than 0.05?

If the p-value is less than the significance level $\alpha =\; 0.05$, it means there is strong evidence against the null hypothesis $H{0}_{}:\beta =0$. Therefore, we reject the null hypothesis and conclude that the slope $\beta$ is significantly different from zero. This indicates a statistically significant linear correlation between the explanatory and response variables.

When should you use a one-tailed versus a two-tailed hypothesis test for the slope?

Use a two-tailed test when you want to check if the slope $\beta$ is simply different from zero, without specifying direction ($H{A}_{}:\beta \ne 0$). Use a one-tailed test when you have a specific direction in mind: if you expect a positive relationship, test $H{A}_{}:\beta >0$; if negative, test $H{A}_{}:\beta <0$. The choice depends on the research question or claim being tested.