High School

An economist is studying inflation in electricity prices in 2018 and 2019. He samples 9 different counties and believes that the average price of electricity, even after adjusting for inflation, has changed. His results are given in the following table. Test the economist's claim about the average price of electricity in each county for each year, adjusting the prices for inflation. Assume that the population distribution of the paired differences is approximately normal. Let prices in 2018 be Population 1 and prices in 2019 be Population 2. Use a significance level of 0.05.

[tex]\[

\begin{array}{|c|c|}

\hline

\multicolumn{2}{|c|}{\text{Average Residential Retail Prices of Electricity (\$/kWh)}} \\

\hline

2018 & 2019 \\

\hline

15.51 & 16.39 \\

\hline

14.19 & 15.92 \\

\hline

18.48 & 20.44 \\

\hline

13.27 & 15.11 \\

\hline

15.83 & 16.44 \\

\hline

18.25 & 19.60 \\

\hline

11.07 & 12.90 \\

\hline

19.77 & 18.57 \\

\hline

12.40 & 14.05 \\

\hline

\end{array}

\][/tex]

Step 1 of 3: State the null and alternative hypotheses for the test.

Fill in the blank below:

[tex]\[

\begin{array}{l}

H_0: \mu_d = 0 \\

H_a: \mu_d \neq 0

\end{array}

\][/tex]

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.