An economist studying inflation in electricity prices in 2018 and 2019 believes that the average price of electricity, even after adjusting for inflation, changed between these two years. To test his claim, he samples 9 different counties and records the average price of electricity in each county from each year. He then adjusts the prices for inflation. His results are given in the following table. Test the economist’s claim at the 0.05 level of significance assuming that the population distribution of the paired differences is approximately normal. Let d= (prices in 2019)−(prices in 2018).
 
Average Residential Retail Prices of Electricity ($/kWh)
2018	2019
19.32	18.19
15.00	13.69
11.20	12.72
14.48	14.51
14.15	12.23
19.17	17.21
13.13	11.29
17.62	15.89
19.06	17.71
 
Step 2 of 3 :  Compute the value of the test statistic. Round your answer to three decimal places.

To test the economist’s claim, we will perform a paired $ t $-test, since we have paired observations from the same counties in both years, and we want to see if there is a significant change in the average electricity prices from 2018 to 2019.

### Step 1: Calculate the differences
We define the differences $ d_i = \text{Price in 2019} - \text{Price in 2018} $ for each county.

Let's calculate these differences:

| 2018 Price ($/kWh) | 2019 Price ($/kWh) | Difference ($ d_i $) |
|--------------------|--------------------|-----------------------|
| 19.32              | 18.19              | $ 18.19 - 19.32 = -1.13 $  |
| 15.00              | 13.69              | $ 13.69 - 15.00 = -1.31 $  |
| 11.20              | 12.72              | $ 12.72 - 11.20 = 1.52 $   |
| 14.48              | 14.51              | $ 14.51 - 14.48 = 0.03 $   |
| 14.15              | 12.23              | $ 12.23 - 14.15 = -1.92 $  |
| 19.17              | 17.21              | $ 17.21 - 19.17 = -1.96 $  |
| 13.13              | 11.29              | $ 11.29 - 13.13 = -1.84 $  |
| 17.62              | 15.89              | $ 15.89 - 17.62 = -1.73 $  |
| 19.06              | 17.71              | $ 17.71 - 19.06 = -1.35 $  |

### Step 2: Calculate the mean and standard deviation of the differences

1. **Mean of the differences, $ \bar{d} $:**
   \[
   \bar{d} = \frac{\sum d_i}{n}
   \]

2. **Standard deviation of the differences, $ s_d $:**
   \[
   s_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n-1}}
   \]

### Step 3: Compute the test statistic $ t $
\[
t = \frac{\bar{d}}{s_d / \sqrt{n}}
\]

Let's calculate these values.

I'll compute these now.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can guide you through the manual calculation steps to find the value of the test statistic, or help with any other related questions. Let me know how you'd like to proceed!

Here are some related questions to consider:
1. How is the paired $ t $-test different from an independent $ t $-test?
2. What are the assumptions of the paired $ t $-test?
3. What does the $ p $-value in a $ t $-test represent?
4. How do you interpret a statistically significant result in this context?
5. What are the limitations of using a paired $ t $-test in real-world data?

**Tip:** In a paired $ t $-test, always check that the differences between paired observations are approximately normally distributed.

An economist studying inflation in electricity prices in 2018 and 2019 believes that the average price of electricity, even after adjusting for inflation, changed between these two years. To test his claim, he samples 9 different counties and records the average price of electricity in each county from each year. He then adjusts the prices for inflation. His results are given in the following table. Test the economist’s claim at the 0.05 level of significance assuming that the population distribution of the paired differences is approximately normal. Let d = (prices in 2019) − (prices in 2018).

This problem demonstrates how to conduct a paired t-test to determine if inflation-adjusted electricity prices changed between 2018 and 2019. Using sample data from 9 counties, the problem explores key statistical concepts such as hypothesis testing, calculating the test statistic, and determining significance at the 0.05 level. Suitable for college-level statistics students.

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