Suppose A Study Reports The Average Price For A Gallon

Suppose A Study Reports That The Average Price For A Gallon Of Self Se

Suppose a study reports that the average price for a gallon of self-serve regular unleaded gasoline is $3.16. You believe that the figure is higher in your area of the country. You decide to test this claim for your part of the United States by randomly calling gasoline stations. Your random survey of 25 stations produces the following prices (all in dollars). Assume gasoline prices for a region are normally distributed.

Did the data you obtained provide enough evidence to reject the claim? Use a 1% level of significance. Make sure you clearly state both the null and the alternative hypotheses in full sentences. Following your calculations, clearly state the conclusion in the same manner (do not simply say “accept/reject null”). Explain how you arrived at this conclusion (based on which metrics).

Paper For Above instruction

The goal of this analysis is to test whether the average gasoline price in a specific region exceeds the reported national average of $3.16. Given the sample data collected from 25 gasoline stations, we will perform a statistical hypothesis test at a significance level of 1% to determine whether the claim can be rejected based on this data.

Formulation of Hypotheses

The null hypothesis (H₀) and the alternative hypothesis (H_a) are set as follows:

• Null hypothesis (H₀): The true mean price of gasoline in the region is equal to $3.16.
• Alternative hypothesis (H_a): The true mean price of gasoline in the region is greater than $3.16.

This is a one-tailed test because the research interest is specifically whether the local average price is higher than the national average.

Data Summary and Calculations

The sample data provided are:

3.27, 3.3, 3.16, 3.15, 3.11, 3.05, 3.08, 3.12, 3.13, 3.14, 3.16, 3.19, 3.27, 3.14, 3.14, 3.2, 3.3, 3.09, 3.05, 3.07, 3.37, 3.34, 3.35, 3.35, 3.1

Firstly, let's compute the sample mean and standard deviation:

• Sample size (n): 25
• Sample mean (x̄):

Calculating Sample Mean

The sum of all observations is:

3.27 + 3.3 + 3.16 + 3.15 + 3.11 + 3.05 + 3.08 + 3.12 + 3.13 + 3.14 + 3.16 + 3.19 + 3.27 + 3.14 + 3.14 + 3.2 + 3.3 + 3.09 + 3.05 + 3.07 + 3.37 + 3.34 + 3.35 + 3.35 + 3.1 = 79.07

Thus, the sample mean:

x̄ = 79.07 / 25 = 3.1628

Next, we calculate the sample standard deviation (s):

Summing the squared deviations:

Sum of squared deviations ≈ 0.357

The sample variance (s²) is:

s2 = 0.357 / (n - 1) = 0.357 / 24 ≈ 0.01488

And the sample standard deviation:

s = √0.01488 ≈ 0.122

Test Statistic Calculation

The standard error (SE) of the mean is:

SE = s / √n = 0.122 / √25 = 0.122 / 5 = 0.0244

Calculate the t-statistic:

t = (x̄ - μ₀) / SE = (3.1628 - 3.16) / 0.0244 ≈ 0.0028 / 0.0244 ≈ 0.115

Decision Making

At a significance level of 1% (α = 0.01), and with 24 degrees of freedom, the critical t-value for a one-tailed test can be obtained from t-distribution tables or software:

• Critical t-value ≈ 2.492

Since the calculated t-statistic (≈ 0.115) is much less than the critical t-value (≈ 2.492), we fail to reject the null hypothesis.

Conclusion

Based on the sample data and the results of the t-test, there is insufficient evidence at the 1% significance level to support the claim that the average price of gasoline in this region exceeds $3.16. The observed sample mean of approximately $3.163 is very close to the national average, and the t-statistic indicates that any difference is likely due to random sampling variability rather than a true difference in population means.

In conclusion, we do not have enough statistical evidence to reject the null hypothesis. Therefore, we cannot conclude that the local average gasoline price is higher than the reported national average of $3.16 at the 1% significance level.

References

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