Final answer:
To test the alternative hypothesis that the median is different from 100 using a sign test at the α=0.05 level, formulate the hypotheses, compile the number of positive and negative signs relative to 100, calculate the test statistic, and then compare it to the critical value or p-value to determine whether to reject the null hypothesis.
Step-by-step explanation:
To test the given alternative hypothesis at the α=0.05 level of significance that the median is different from 100, we use the sign test. This is a non-parametric test that does not require the assumption of normal distributions. The procedure would be:
- Formulate the null hypothesis (⅐) that the median is equal to 100 and the alternative hypothesis (⅔) that the median is not equal to 100.
- Count the number of sample observations that are above and below the hypothesized median of 100.
- Calculate the test statistic based on the smaller of the counts of signs (either '+' or '-').
- Determine the critical value or p-value from the binomial distribution with n equal to the total number of non-ties and p=0.5.
- Compare the test statistic to the critical value, or compare the p-value to α to decide whether to reject the null hypothesis.
If the p-value is less than α, we reject the null hypothesis and conclude that there is enough evidence to suggest the median is different from 100. Otherwise, we fail to reject the null hypothesis.