In prior investigations made by a local Gas and Electric Company, 65% of homeowners prefer to heat with gas instead of electricity. In a recent study of 200 homeowners 60% prefer heating with gas. In a two tailed test at a .05 level of significance can we conclude that the percentage of homeowners who prefer gas differs from the previously reported percentage of 65%?
What is the Null Hypotheses Ho?
What is the Alternate Hypotheses H1?
Sketch the Normal Distribution and Standard Normal Distribution for the Ho hypothesis. Identify the reject region(s) and show the critical value(s).
(Clearly delineate in each region what the decision would be.)
What is the decision rule?
What is the value of the Test Statistic? (Calculate answer and show equation with numerical numbers.) (The value of the test statistic is the numeric value of z or t calculated from the data.)
Based upon the data, what is the decision Ho or H1?
Based upon your decision, what are the two possible outcomes for your decision? (Type I, Type II, Correct)
To answer these questions, we need to run a hypothesis test. The hypothesis test will help us determine if the percentage of homeowners who prefer gas heating differs from the previously reported percentage of 65%.
1. Null Hypotheses Ho:
The null hypothesis states that there is no difference in the percentage of homeowners who prefer gas heating compared to the previously reported percentage of 65%.
Ho: p = 0.65
2. Alternate Hypotheses H1:
The alternate hypothesis states that there is a difference in the percentage of homeowners who prefer gas heating compared to the previously reported percentage of 65%.
H1: p ≠ 0.65 (two-tailed test)
To sketch the normal distribution and standard normal distribution for the Ho hypothesis, we need the mean and standard deviation. In this case, we don't have the raw data, but only the proportions.
Let's calculate the sample proportion (p̂) from the recent study:
p̂ = X/n = 0.60 (where X is the number of homeowners who prefer gas and n is the sample size of 200)
The distribution of sample proportions can be approximated by a normal distribution when the sample size is large (n ≥ 30). We assume that the sampling distribution follows a normal distribution.
Now we can calculate the standard error:
SE = sqrt((p̂ * (1 - p̂)) / n)
SE = sqrt((0.60 * (1 - 0.60)) / 200)
SE = 0.035
The test statistic for this hypothesis test is a Z-statistic. We can calculate it using the formula:
Z = (p̂ - p0) / SE
Z = (0.60 - 0.65) / 0.035
Z = -1.428
To determine the decision rule, we need to establish the critical value(s) at the 0.05 level of significance for a two-tailed test. Since this is a two-tailed test, we will split the significance level evenly into the two critical regions.
The critical value for the lower tail is -1.96, and the critical value for the upper tail is +1.96. These critical values define the reject regions.
Decision Rule:
If the calculated Z-statistic falls outside the critical value range, we will reject the null hypothesis. If the calculated Z-statistic falls within the critical value range, we will fail to reject the null hypothesis.
The calculated Z-statistic is -1.428. Since -1.428 falls within the critical value range of -1.96 to +1.96, we fail to reject the null hypothesis.
Based on the decision, we fail to reject the null hypothesis Ho: p = 0.65. Therefore, we do not have enough evidence to conclude that the percentage of homeowners who prefer gas differs from the previously reported percentage of 65%.
The two possible outcomes for our decision are Correct and Type II error:
- Correct decision: We correctly fail to reject the null hypothesis when it is true (Ho: p = 0.65).
- Type II Error: We fail to reject the null hypothesis when it is false (Ho: p ≠ 0.65). In other words, we fail to detect a difference in the percentages of homeowners who prefer gas heating.