Question

It was reported that last year the average price of a gallon gasoline in a city X was $3.15. This year a sample of 50 gas station had an average price of $3.10 for a gallon. We assume that the population standard deviation of prices is $0.15. We are interested in determining whether this year mean price is less than last year. Perform a hypothesis test at the level of significance α=0.05.The assumptions are satisfied:

It is a simple random sample The sample size is
so it is more than required number 30.

Answer 1

In order to determine if the average price of gasoline this year is less than the average price last year, we will conduct a hypothesis test using the following steps:

### Step 1: Define the null and alternative hypotheses

- **Null hypothesis (H0)**: μ = 3.15 (The average price of gasoline this year is equal to last year)

- **Alternative hypothesis (H1)**: μ < 3.15 (The average price of gasoline this year is less than last year)

### Step 2: Set the significance level

The significance level (α) is given as 0.05.

### Step 3: Test statistic and calculation

We will use a z-test for this hypothesis test because the population standard deviation is known and the sample size is large (n > 30).

The formula for the z-test statistic is:

\[
z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}
\]

Where:
- \(\bar{x} = 3.10\) (sample mean)
- \(\mu_0 = 3.15\) (population mean under the null hypothesis)
- \(\sigma = 0.15\) (population standard deviation)
- \(n = 50\) (sample size)

Now, plug in the values:

\[
z = \frac{3.10 - 3.15}{\frac{0.15}{\sqrt{50}}} = \frac{-0.05}{\frac{0.15}{7.071}} = \frac{-0.05}{0.021213} \approx -2.36
\]

### Step 4: Determine the critical value

For a one-tailed test at the 0.05 significance level, we'll look up the critical z-value in a z-table or use a calculator. The critical z-value for a left-tailed test (α = 0.05) is approximately:

- **Critical z-value:** -1.645

### Step 5: Make a decision

Now, we compare the calculated z-value to the critical z-value:

- **Calculated z-value:** -2.36
- **Critical z-value:** -1.645

Since -2.36 < -1.645, we reject the null hypothesis.

### Conclusion

There is sufficient evidence to support the claim that the average price of gasoline this year is less than last year's average price of $3.15 at the 0.05 significance level.