The Suva Energy Information Administration reported that 51.7% of homes in Suva were heated by natural gas. A random sample of 200 homes found that 115 were heated by natural gas. Does the evidence supports the claim, or has the percentage changed? (use a P-value method)
To determine whether the evidence supports the claim or if the percentage has changed, we can use a hypothesis test with the P-value method.
Step 1: State the hypotheses.
- Null Hypothesis (H0): The percentage of homes heated by natural gas in Suva is still 51.7%.
- Alternative Hypothesis (Ha): The percentage of homes heated by natural gas in Suva has changed.
Step 2: Set the significance level (alpha).
Let's assume a significance level (alpha) of 0.05, which is a common choice in hypothesis testing.
Step 3: Collect sample data and compute the test statistic.
Given in the question, we have a random sample of 200 homes, with 115 of them heated by natural gas. To compute the test statistic, we need to calculate the z-score.
Sample proportion: p̂ = 115/200 = 0.575 (rounded to three decimal places)
Under the null hypothesis, the expected proportion is 0.517 (51.7%).
Standard error (SE) of the sample proportion: SE = sqrt(p(1-p)/n)
SE = sqrt(0.517(1-0.517)/200) ≈ 0.029
Z-score: z = (p̂ - p) / SE
z = (0.575 - 0.517) / 0.029 ≈ 1.966 (rounded to three decimal places)
Step 4: Calculate the P-value.
The P-value is the probability of obtaining a test statistic as extreme or more extreme than the observed value, assuming the null hypothesis is true. We will calculate the P-value using a standard normal distribution table or statistical software.
P-value ≈ P(Z > 1.966)
Step 5: Make a decision.
Compare the P-value to the significance level (alpha). If the P-value is less than or equal to alpha, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, we would compare the P-value to 0.05. If the calculated P-value is less than 0.05, we reject the null hypothesis and conclude that the evidence supports the claim that the percentage of homes heated by natural gas in Suva has changed. If the P-value is greater than 0.05, we fail to reject the null hypothesis, suggesting that the evidence does not support a change in the percentage.
Please note that I cannot calculate the exact P-value for you without the precise areas under the standard normal distribution curve. You would need to look up the corresponding probability from a standard normal distribution table or use statistical software to get the exact P-value.
To determine if the evidence supports the claim or if the percentage has changed, we can perform a hypothesis test using the P-value method.
Let's define our hypotheses:
- Null hypothesis (H₀): The percentage of homes heated by natural gas in Suva has not changed, i.e., p = 0.517.
- Alternative hypothesis (H₁): The percentage of homes heated by natural gas in Suva has changed, i.e., p ≠ 0.517.
Next, we can calculate the test statistic and the P-value.
1. Test statistic:
To determine if the observed data significantly deviates from the expected proportion (51.7%), we can use the test statistic for proportions, which follows an approximately normal distribution for large enough sample sizes.
The test statistic is calculated using the formula:
t = (p̂ - p₀) / √(p₀(1 - p₀) / n),
where p̂ is the sample proportion, p₀ is the hypothesized population proportion, and n is the sample size.
Given:
- p̂ = 115/200 = 0.575 (sample proportion)
- p₀ = 0.517 (hypothesized population proportion)
- n = 200 (sample size)
Plugging in these values, we get:
t = (0.575 - 0.517) / √(0.517(1 - 0.517) / 200) ≈ 2.004
2. P-value:
The P-value is the probability of observing a test statistic as extreme as the one calculated if the null hypothesis is true.
Using the test statistic, we can find the P-value by comparing it to the appropriate distribution. In this case, we will use the standard normal distribution.
Since the alternative hypothesis is two-sided (p ≠ 0.517), we need to find the area in both tails of the distribution that is as extreme or more extreme than the observed test statistic.
Looking up the cumulative probability in a standard normal distribution table for a test statistic of 2.004 (in both tails), we find a P-value of approximately 0.045.
3. Interpretation:
The P-value of 0.045 is less than the commonly used significance level of 0.05. Therefore, we have evidence to reject the null hypothesis.
Conclusion:
The evidence supports the claim that the percentage of homes in Suva heated by natural gas has changed since it deviates significantly from the reported percentage of 51.7%. The new proportion is approximately 57.5% (based on the sample data).