A manufacterer of computers designed to aid social scientists in analyzing rsearch data claims that are operational for at least 80% of the time . during the course of the year one computer was operational for 270 days. test, at the 1% CONFIDENCE LEVEL, whether the manufacturer's claim was justified. use the five step model clearly showing all steps.
To test whether the manufacturer's claim is justified at the 1% confidence level, you can use a hypothesis test. Here is the step-by-step process for conducting the test:
Step 1: State the hypotheses:
- Null hypothesis (H0): The computer is operational for at least 80% of the time.
- Alternative hypothesis (Ha): The computer is not operational for at least 80% of the time.
Step 2: Determine the significance level:
In this case, the significance level is given as 1%, which means we want to have strong evidence to reject the null hypothesis.
Step 3: Collect the data:
The computer was operational for 270 days during the year.
Step 4: Calculate the test statistic:
We need to calculate a test statistic to compare it against a critical value from the statistical table. In this case, we will use a one-proportion z-test.
To calculate the test statistic, we need to find the sample proportion (p-hat) and the standard error (SE) of the sample proportion. The formula is:
p-hat = x/n
SE = sqrt(p-hat * (1 - p-hat) / n)
Where:
- x is the number of successes (operational days)
- n is the sample size (365 days in a year)
- sqrt refers to the square root
Plugging in the values:
p-hat = 270/365 ≈ 0.7397
SE = sqrt(0.7397 * (1 - 0.7397) / 365) ≈ 0.0191
Step 5: Calculate the critical value and make a decision:
At the 1% confidence level, the critical value is found using a z-distribution table. For a two-tailed test, the critical z-score is ±2.576.
To find the z-score, we use the formula: z = (p - P) / SE
Where:
- p is the hypothesized proportion (80% or 0.8)
- P is the sample proportion (p-hat)
- SE is the standard error
Plugging in the values:
z = (0.8 - 0.7397) / 0.0191 ≈ 3.152
Since the calculated z-score (3.152) is greater than the critical z-score (±2.576), we reject the null hypothesis. This means that there is strong evidence to suggest that the computer is not operational for at least 80% of the time, and the manufacturer's claim is not justified.
Note: The z-score tells us how many standard deviations away from the mean the data point (sample proportion) is. If it falls outside the critical range, it suggests that the observed result is significantly different from the expected result, leading to the rejection of the null hypothesis.