The Acme Car Company claims that at most 8% of its new cars have a manufacturing defect. A quality control inspector randomly selects 300 new cars and finds that 33 have a defect. Should she reject the 8% claim? Assume that the significance level is 0.05.
0.028
To determine if the quality control inspector should reject the 8% claim made by the Acme Car Company, we can perform a hypothesis test. Here's how you can go about it:
Step 1: Define the null and alternative hypothesis:
- Null hypothesis (H0): The percentage of new cars with a manufacturing defect is equal to or less than 8%. (p <= 0.08)
- Alternative hypothesis (Ha): The percentage of new cars with a manufacturing defect is greater than 8%. (p > 0.08)
Step 2: Determine the test statistic and the rejection region:
In this case, we'll use the binomial distribution to calculate the probability of observing 33 or more defective cars out of a sample of 300, assuming the null hypothesis is true. We can then compare this probability with the significance level of 0.05.
Step 3: Calculate the p-value:
The p-value represents the probability of observing a sample result as extreme or more extreme than what was actually observed, assuming the null hypothesis is true. In this case, we need to calculate the probability of observing 33 or more defective cars out of 300, assuming that the true percentage of defects is 8% or less.
Using statistical software or a binomial distribution calculator, we find that the probability of observing 33 or more defective cars out of 300, assuming p <= 0.08, is approximately 0.037.
Step 4: Compare the p-value with the significance level:
If the p-value is less than the significance level (0.05 in this case), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, the calculated p-value of 0.037 is less than the significance level of 0.05. Therefore, we reject the null hypothesis and conclude that there is sufficient evidence to suggest that the percentage of new cars with a manufacturing defect is higher than 8%.
In simpler terms, the quality control inspector should reject the Acme Car Company's claim of at most 8% defects in their new cars based on the sample data.
To determine whether the quality control inspector should reject the 8% claim, we can perform a hypothesis test using a significance level of 0.05.
Let's set up the null and alternative hypotheses:
Null hypothesis (H0): The proportion of new cars with a manufacturing defect is equal to 8%.
Alternative hypothesis (Ha): The proportion of new cars with a manufacturing defect is not equal to 8%.
Using the given information, we can calculate the sample proportion and use it to perform a hypothesis test.
Sample proportion (p̂) = Number of defective cars / Total number of cars sampled = 33/300 = 0.11
To perform the hypothesis test, we can use the z-test for proportions. The test statistic formula is:
z = (p̂ - P) / sqrt(P*(1-P)/n)
Where:
- P is the hypothesized proportion (8% or 0.08 in this case)
- n is the sample size (300 in this case)
Calculating the test statistic:
z = (0.11 - 0.08) / sqrt(0.08*(1-0.08)/300) = 1.23
Next, we need to find the critical value for a two-tailed test at the 0.05 significance level. Since we're using a significance level of 0.05, we need to divide it by 2 to get 0.025 for each tail.
Using a standard normal distribution table or a calculator, we find that the critical z-value for a 0.025 significance level is approximately ±1.96.
Since the calculated z-value of 1.23 is within the range of -1.96 to +1.96, we fail to reject the null hypothesis.
Therefore, there is not enough evidence to reject the claim that at most 8% of Acme Car Company's new cars have a manufacturing defect.