The Suva Energy Information Administration reported that 51.7% of homes in Sunnyville 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).

10

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 support the claim? (Use   0.05 and a P-value method).

To determine whether the evidence supports the claim or if the percentage has changed, we need to conduct a hypothesis test using the P-value method.

Let's define the null and alternative hypotheses:

Null hypothesis (H0): The percentage of homes in Sunnyville heated by natural gas is 51.7% (no change).
Alternative hypothesis (Ha): The percentage of homes in Sunnyville heated by natural gas has changed.

Next, we need to calculate the test statistic and the P-value. In this case, the test statistic will follow a normal distribution because the sample size is large (n=200) and the data is based on proportions.

The test statistic can be calculated using the formula:

z = (p - P) / sqrt(P * (1 - P) / n)

Where:
p = sample proportion (115/200 = 0.575)
P = hypothesized proportion (0.517)
n = sample size (200)

Calculating the test statistic:

z = (0.575 - 0.517) / sqrt(0.517 * (1 - 0.517) / 200)
z = (0.058) / sqrt(0.517 * 0.483 / 200)
z = 0.058 / sqrt(0.249411 / 200)
z = 0.058 / sqrt(0.00124705)
z ≈ 0.058 / 0.03530296
z ≈ 1.641

To find the P-value associated with z = 1.641, we consult the standard normal distribution table or use statistical software. The P-value is the probability of observing a test statistic as extreme as (or more extreme than) the one calculated under the assumption that the null hypothesis is true.

By looking up the P-value in the standard normal distribution table, we find that the P-value for z = 1.641 is approximately 0.0507.

Since the P-value (0.0507) is greater than the commonly accepted significance level of 0.05 (assuming α = 0.05), we fail to reject the null hypothesis.

Therefore, based on the P-value method, there is not enough evidence to support the claim that the percentage of homes in Sunnyville heated by natural gas has changed.

To determine whether the evidence supports the claim or the percentage has changed, we can use a hypothesis test with the P-value method. The null hypothesis (H0) assumes that the percentage of homes heated by natural gas is still 51.7%, and the alternative hypothesis (Ha) assumes that the percentage has changed.

Let's go step by step on how to conduct the hypothesis test:

Step 1: Define the hypotheses.
H0: The percentage of homes heated by natural gas is 51.7%.
Ha: The percentage of homes heated by natural gas has changed.

Step 2: Determine the test statistic.
In this case, we will use the test statistic for a hypothesis test for proportions. The formula for the test statistic is:

z = (p̂ - p0) / √(p0 * (1-p0) / n)

where:
p̂ is the sample proportion (115/200 = 0.575),
p0 is the assumed proportion under the null hypothesis (0.517),
and n is the sample size (200).

Calculating these values, we get:
z = (0.575 - 0.517) / √(0.517 * (1-0.517) / 200) = 1.96 (rounded to two decimal places)

Step 3: Determine the P-value.
The P-value is the probability of obtaining a test statistic as extreme or more extreme than the observed test statistic, assuming the null hypothesis is true. In other words, it represents the evidence against the null hypothesis.

To find the P-value, we need to calculate the probability of getting a z-score as large as 1.96 or more extreme, assuming a standard normal distribution. Looking up the z-value in a standard normal distribution table, we find that the corresponding P-value is approximately 0.025.

Step 4: Draw a conclusion.
If the P-value is less than the significance level (α), usually set to 0.05, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

In this case, the P-value is 0.025, which is less than the significance level. Therefore, we reject the null hypothesis. The evidence supports the claim that the percentage of homes heated by natural gas has changed.

Please note that this conclusion assumes that the sample is representative and independent of the population.