à manufacturer of computers designed to aid social scientists in analyzing research data clams that his computers are opérational for at least 80 percent of the time. during the course of the year one computer was operational for 270 days. test, 1 percent confidence level, whether the manufacturer's claim was justified. use the rive step model clearly showing all steps.
You can try a proportional one-sample z-test for this problem. Check a table for the critical value to compare to the test statistic. If the test statistic exceeds the critical value, then reject the null and accept the alternative hypothesis.
To test the manufacturer's claim at a 1 percent confidence level, we need to follow a five-step hypothesis testing process. Here are the steps:
Step 1: State the hypotheses:
- Null hypothesis (H0): The manufacturer's claim is justified. The computers are operational for at least 80 percent of the time.
- Alternative hypothesis (Ha): The manufacturer's claim is not justified. The computers are operational for less than 80 percent of the time.
Step 2: Set the significance level (alpha):
In this case, the significance level is given as 1 percent, which means alpha = 0.01.
Step 3: Compute the test statistic and the critical value:
We'll analyze the sample data to calculate the test statistic and then compare it with the critical value from the appropriate statistical distribution.
Let's define "success" as a day when the computer is operational. Since we're dealing with proportions, we'll use a one-sample proportion z-test. The formula for the test statistic (z) is:
z = (p - P0) / sqrt((P0 * (1 - P0)) / n)
where:
p is the sample proportion (observed proportion of operational days)
P0 is the hypothesized proportion (in this case, 0.8)
n is the sample size (number of days observed)
In this case, p = 270/365 = approximately 0.7397, P0 = 0.8, n = 365.
Plugging these values into the formula, we can calculate the test statistic:
z = (0.7397 - 0.8) / sqrt((0.8 * (1 - 0.8)) / 365)
Step 4: Determine the critical value:
Since the alternative hypothesis states that the computers are operational for less than 80 percent of the time, this is a lower-tailed test. We'll find the critical value from the standard normal distribution table (z-table) corresponding to a 1 percent significance level (alpha = 0.01).
Step 5: Make a decision:
If the test statistic falls within the critical region (i.e., it is less than the critical value), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
To summarize, we need to calculate the test statistic (z), find the critical value from the z-table, and compare the two to make a decision about rejecting or failing to reject the null hypothesis.