A pollster conducts a survey of a random sample of voters in a community to estimate the proportion who support a measure on public health insurance. Let P be the proportion of the population who support the measure. The pollster takes the sample of 200 voters and tests Ho: P=0.5 versus H1:P(not equal)0.5 at 5% level. What is the power of the test if the true value of P is 0.55?
The probability of making a Type II error is equal to beta. A Type II error is failure to reject the null when it is false. The power of the test is 1-beta and is the correct decision of rejecting the null when it is false. The alpha level directly affects the power of a test. The higher the level, the more powerful the test. Sample size also affects power. If your question is asking to calculate the actual power of the test, you will need to find methods to calculate beta.
I hope this helps.
To calculate the power of the test, we need to determine the probability of rejecting the null hypothesis when the alternative hypothesis is true. In this case, the null hypothesis (Ho) is that the proportion of voters who support the measure is 0.5, and the alternative hypothesis (H1) is that the proportion is not equal to 0.5.
To find the power of the test, we need the sample size (n), the significance level (α), and the true proportion (P).
Given:
Sample size (n) = 200
Significance level (α) = 5% = 0.05
True proportion (P) = 0.55
The power of the test can be calculated using statistical software or by referring to a table of the standard normal distribution. Since the calculations are complex, I will explain the general steps involved.
1. Calculate the standard error (SE) of the sample proportion using the formula:
SE = sqrt((P * (1 - P)) / n)
For the given values: SE = sqrt((0.55 * (1 - 0.55)) / 200)
2. Determine the critical values (z) for the rejection regions based on the significance level. Since the alternative hypothesis is two-sided, we divide the significance level by 2 and find the critical values from the standard normal distribution. In this case, the critical values are -1.96 and 1.96 (for a 5% significance level).
3. Calculate the standardized test statistic (z):
z = (sample proportion - hypothesized proportion) / SE
For the given values: z = (0.55 - 0.5) / SE
4. Calculate the power of the test by determining the probability of obtaining a test statistic outside the rejection region, given the true proportion:
Power = P(z < -1.96 or z > 1.96 | true proportion)
This probability can be calculated using the standard normal distribution or statistical software.
You can use online calculators or statistical software (such as R or Python) to calculate the power of the test using the given values. Just input the necessary parameters, including the sample size, significance level, hypothesized proportion, and true proportion. The software will provide you with the power of the test.