suppose i guess that the average income of a family is $40,000. Display a test of the hypothesis that the mean of the numerical variable equals the value of the guess given versus the alternative that it does not equal this value. then what would be my decision at the 5% significance level
To test the hypothesis that the mean income of a family is $40,000, you can follow these steps:
Step 1: Define the null and alternative hypotheses:
- Null hypothesis (H0): The mean income of a family is $40,000.
- Alternative hypothesis (H1): The mean income of a family is not equal to $40,000.
Step 2: Collect a representative sample:
- Randomly select a sample of families and collect their income data. Ensure that the sample is representative of the population you are interested in.
Step 3: Calculate the sample mean and sample standard deviation:
- Calculate the mean of the income values in your sample.
- Calculate the standard deviation of the income values in your sample.
Step 4: Perform a t-test:
- Use a t-test to determine whether the sample mean differs significantly from the hypothesized mean of $40,000.
- The t-test will tell you the likelihood that the observed difference between the sample mean and hypothesized mean is due to random chance.
Step 5: Determine the p-value and make a decision:
- The p-value is the probability of observing a sample mean as extreme as the one you found, assuming the null hypothesis is true.
- Compare the p-value to the significance level (in this case, 5% or 0.05).
- If the p-value is less than the significance level (p < 0.05), reject the null hypothesis. This means that the mean income of families is not equal to $40,000.
- If the p-value is greater than or equal to the significance level (p ≥ 0.05), fail to reject the null hypothesis. This means that there is not enough evidence to conclude that the mean income of families is different from $40,000.
It's important to note that the decision to reject or fail to reject the null hypothesis is based on statistical evidence and not definitive proof.