Answer :
To solve this question, we'll be setting up the null and alternative hypotheses for a paired t-test. A paired t-test is used when we are comparing two related groups, in this case, the electricity prices across two years for the same counties. Here's how we'll set that up:
1. Null Hypothesis (H₀): The null hypothesis assumes that there is no difference in the average electricity prices between the two years. This can be mathematically represented as:
- [tex]\( H_0: \mu_d = 0 \)[/tex]
- This means the mean of the differences ([tex]\(\mu_d\)[/tex]) between the 2018 and 2019 electricity prices is zero, implying no change.
2. Alternative Hypothesis (Hₐ): The alternative hypothesis suggests that there is a difference in the average electricity prices between the two years. Mathematically, this is written as:
- [tex]\( H_a: \mu_d \neq 0 \)[/tex]
- This indicates that the mean of the differences ([tex]\(\mu_d\)[/tex]) is not zero, meaning there has been a change in prices.
The hypotheses set up in this way will allow us to perform a paired t-test to determine if the changes in electricity prices are statistically significant. This approach checks if the average difference is greater than what would be expected by random chance, based on the sample data provided.
1. Null Hypothesis (H₀): The null hypothesis assumes that there is no difference in the average electricity prices between the two years. This can be mathematically represented as:
- [tex]\( H_0: \mu_d = 0 \)[/tex]
- This means the mean of the differences ([tex]\(\mu_d\)[/tex]) between the 2018 and 2019 electricity prices is zero, implying no change.
2. Alternative Hypothesis (Hₐ): The alternative hypothesis suggests that there is a difference in the average electricity prices between the two years. Mathematically, this is written as:
- [tex]\( H_a: \mu_d \neq 0 \)[/tex]
- This indicates that the mean of the differences ([tex]\(\mu_d\)[/tex]) is not zero, meaning there has been a change in prices.
The hypotheses set up in this way will allow us to perform a paired t-test to determine if the changes in electricity prices are statistically significant. This approach checks if the average difference is greater than what would be expected by random chance, based on the sample data provided.
