t has been claimed that 65% of homeowners
would prefer to heat with electricity instead of gas.
A study finds that 60% of 2000 homeowners
prefer electric heating to gas. Can we conclude
that the percentage who prefer electric heating
differs from 65%? Use alpha = 10%
To determine whether the percentage of homeowners who prefer electric heating differs from 65%, we can conduct a hypothesis test.
Hypotheses:
- Null Hypothesis (H0): The percentage of homeowners who prefer electric heating is equal to 65%.
- Alternative Hypothesis (Ha): The percentage of homeowners who prefer electric heating is different from 65%.
Test Statistic:
We will use the z-test for proportions to compare the observed proportion (60% of 2000 homeowners) with the hypothesized proportion (65%).
Assumptions:
- The sample is random and representative of the population.
- The sample size is sufficiently large (n > 30).
Calculation:
1. Calculate the standard error (SE) for the proportion:
SE = √[(p * (1 - p)) / n]
where p is the hypothesized proportion (65%) and n is the sample size (2000).
2. Calculate the z-value:
z = (observed proportion - hypothesized proportion) / SE
3. Determine the critical z-values for a two-tailed test with a significance level of 10%.
The critical z-values can be obtained from the standard normal distribution or a z-table.
4. Compare the calculated z-value with the critical z-values:
- If the calculated z-value is outside the critical z-values, reject the null hypothesis.
- If the calculated z-value is within the critical z-values, fail to reject the null hypothesis.
Decision:
If we reject the null hypothesis, we can conclude that the percentage of homeowners who prefer electric heating differs from 65%. If we fail to reject the null hypothesis, we cannot conclude that the percentage differs.
Note: Since the question does not specify the proportion of the population who prefer electric heating, we are assuming the 65% given in the claim as the hypothesized proportion.
To determine whether we can conclude that the percentage of homeowners who prefer electric heating differs from 65%, we can perform a hypothesis test.
1. State the null hypothesis (H0) and the alternative hypothesis (Ha):
- Null hypothesis (H0): The percentage of homeowners who prefer electric heating is equal to 65%.
- Alternative hypothesis (Ha): The percentage of homeowners who prefer electric heating is not equal to 65%.
2. Set the significance level, alpha, to 10%. This means that there is a 10% chance of rejecting the null hypothesis when it is actually true.
3. Calculate the test statistic:
- In this case, we will use the z-test for proportions since we have information about the percentage of homeowners who prefer electric heating.
- The formula for the z-test for proportions is:
z = (p - P) / sqrt(PQ/n), where
- p is the sample proportion (60% or 0.60 in this case)
- P is the hypothesized population proportion (65% or 0.65 in this case)
- Q is the complement of P (1 - P)
- n is the sample size (2000 in this case)
- Calculating the values:
- p = 0.60
- P = 0.65
- Q = 1 - P = 1 - 0.65 = 0.35
- n = 2000
- Plugging the values in the formula:
z = (0.60 - 0.65) / sqrt((0.65 * 0.35) / 2000)
4. Determine the critical value:
- Since the alternative hypothesis is two-tailed (not equal to), we need to consider both tails of the distribution.
- To find the critical value, we divide the significance level (alpha) by two and find the corresponding z-score from the standard normal distribution table.
- In this case, alpha = 0.10, so alpha/2 = 0.05.
- From the standard normal distribution table, the z-score corresponding to a cumulative probability of 0.05 in each tail is approximated to be 1.96.
5. Compare the test statistic with the critical value:
- If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis.
- If the absolute value of the test statistic is less than or equal to the critical value, we fail to reject the null hypothesis.
6. Calculate the test statistic value:
- Plugging the values in the formula:
z = (0.60 - 0.65) / sqrt((0.65 * 0.35) / 2000)
z ≈ (-0.05) / sqrt((0.65 * 0.35) / 2000)
z ≈ (-0.05) / sqrt(0.2275 / 2000)
z ≈ (-0.05) / sqrt(0.00011375)
z ≈ (-0.05) / 0.0106652
z ≈ -4.684
7. Compare the test statistic with the critical value:
- Since the absolute value of the test statistic (-4.684) is greater than the critical value (1.96), we reject the null hypothesis.
8. Conclusion:
- Based on the results of the hypothesis test, we can conclude that there is sufficient evidence to suggest that the percentage of homeowners who prefer electric heating differs from 65% at the 10% significance level (alpha = 10%).