Problem 7 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

In a study about the effect of wall insulation, the weekly gas consumption (in 1000 cubic feet) and the average outside temperature (in degrees Celsius) was measured of a certain house in southeast England, for 26 weeks before and 30 weeks after cavity-wall insulation had been installed. The house thermostat was set at 20 degrees throughout. The data are listed in Table 27.7. We model the data before insulation by means of a simple linear regression model with normally distributed errors and gas consumption as response variable. A similar model was used for the data after insulation. Given are Before insulation: \(\hat{\alpha}=6.8538, \hat{\beta}=-0.3932\) and \(s_{a}=0.1184, s_{b}=0.0196\) After insulation: \(\quad \hat{\alpha}=4.7238, \hat{\beta}=-0.2779\) and \(s_{a}=0.1297, s_{b}=0.0252\). a. Use the data before insulation to investigate whether smaller outside temperatures lead to higher gas consumption. Formulate the proper null and alternative hypothesis, compute the value of the test statistic, and report your conclusion, using significance level \(0.05\). b. Do the same for the data after insulation.

Step-by-Step Solution

Answer

Both before and after insulation, smaller outside temperatures lead to higher gas consumption.

Step 1: Formulate the Hypotheses Before Insulation

We want to test if smaller outside temperatures lead to higher gas consumption, which translates to testing if the slope (\(\beta\)) is less than 0. \[ H_0: \beta = 0 \H_a: \beta < 0 \] The null hypothesis \(H_0\) states that there is no relationship between outside temperature and gas consumption, while the alternative hypothesis \(H_a\) states that there is a negative relationship.

Step 2: Calculate the Test Statistic Before Insulation

With \(\hat{\beta} = -0.3932\) and standard error \(s_b = 0.0196\), the test statistic \(t\) can be computed as: \[ t = \frac{(\hat{\beta} - 0)}{s_b} = \frac{-0.3932}{0.0196} = -20.06 \] This statistic follows a t-distribution with \(n-2\) degrees of freedom, where \(n\) is the number of observations before insulation.

Step 3: Compare Test Statistic to Critical Value Before Insulation

For a significance level of 0.05 and degrees of freedom of \(24\) (since 26 observations minus 2), we find the critical value from the t-distribution table, which for a one-tailed test is approximately -1.711. Since the computed \(t\) statistic of -20.06 is less than -1.711, we reject \(H_0\).

Step 4: Conclusion for Before Insulation

Since we rejected the null hypothesis, we conclude that there is significant evidence to suggest that smaller outside temperatures do lead to higher gas consumption before insulation.

Step 5: Formulate the Hypotheses After Insulation

The hypotheses remain the same after insulation:\[ H_0: \beta = 0 \H_a: \beta < 0 \] We test whether there is a negative relationship between temperature and gas consumption after insulation.

Step 6: Calculate the Test Statistic After Insulation

With \(\hat{\beta} = -0.2779\) and standard error \(s_b = 0.0252\), the test statistic \(t\) is calculated as: \[ t = \frac{(\hat{\beta} - 0)}{s_b} = \frac{-0.2779}{0.0252} = -11.03 \] This also follows a t-distribution, but with \(28\) degrees of freedom after insulation (since 30 observations minus 2).

Step 7: Compare Test Statistic to Critical Value After Insulation

Using a significance level of 0.05 and degrees of freedom of \(28\), the critical value for a one-tailed test is approximately -1.701. Since the \(t\) statistic of -11.03 is less than -1.701, we reject \(H_0\).

Step 8: Conclusion for After Insulation

We conclude that even after insulation, there is significant evidence to suggest that smaller outside temperatures lead to higher gas consumption.

Key Concepts

Hypothesis Testing

Hypothesis testing is a fundamental part of statistical analysis. It allows us to make inferences or draw conclusions from data by testing predefined assumptions. Typically, this process involves stating two hypotheses: the null hypothesis

• The null hypothesis, denoted as \(H_0\), represents a statement of no effect or no difference. It posits that there is no relationship between the variables being studied.
• The alternative hypothesis, denoted as \(H_a\), is a statement that suggests a potential effect or difference. It indicates that the variables may indeed be related.

Before conducting a hypothesis test, you must determine these hypotheses based on the research question. In our case, we're examining if smaller outside temperatures lead to higher gas consumption. Thus, we test:

• \(H_0: \beta = 0\), asserting no relationship between temperature and consumption.
• \(H_a: \beta < 0\), suggesting that gas consumption increases as temperature decreases.

The essence of hypothesis testing is to assess whether the evidence (data) strongly supports rejecting the null hypothesis in favor of the alternative. If enough evidence exists, the null hypothesis is rejected, leading us to conclude that the alternative is likely true.

t-Distribution

When performing a hypothesis test, especially in linear regression, the test statistic is crucial in determining the result. Often, this statistic follows a specific distribution, known as the t-distribution. This mathematical distribution is commonly used when working with small sample sizes, giving a sense of how likely certain outcomes are. The test statistic, in our case, is calculated using the formula: \[ t = \frac{(\hat{\beta} - 0)}{s_b} \]Here, \(\hat{\beta}\) is the estimated coefficient (slope), and \(s_b\) is the standard error of this coefficient. The t-statistic measures how many standard deviations the estimated coefficient is from zero, under the assumption that \(\beta = 0\).

• For each observation set, the degrees of freedom for the t-distribution are calculated as the number of observations minus two (since we estimate two parameters, \(\alpha\) and \(\beta\)).
• The t-distribution is symmetric and approaches the normal distribution as the sample size increases.

Knowing the degrees of freedom and the calculated t-statistic helps in comparing this value to a critical value, aiding in making informed conclusions on hypothesized relationships.

Significance Level

A significance level in hypothesis testing is a threshold that determines whether the observed result is statistically significant. It represents the probability of rejecting the null hypothesis when it is actually true, which is also known as a Type I error. Typically, the significance level is denoted by \(\alpha\) and is expressed as a percentage (often set at 5%, or 0.05). This means there's a 5% risk of concluding that a relationship between variables exists when it doesn't. To apply this, follow these simple steps:

• Calculate the test statistic and determine the degrees of freedom for your data.
• Locate the critical value from the relevant t-distribution table using the significance level and degrees of freedom.
• Compare the test statistic against the critical t-value.

If the calculated t-statistic falls in the critical region (beyond the critical value in either direction for a two-tailed test, or in the direction of interest for a one-tailed test), then we reject the null hypothesis. In our case, we compared the test statistics with critical values at \(\alpha = 0.05\), finding significant evidence against the null hypothesis.