b) Distinguish between Type I and Type II error. The breaking strengths of cables produced by a manufacturer have mean 1800 lb and standard deviation 100 lb. By a new technique in the manufacturing process, it is claimed that the breaking strength can be increased. To test this claim, a sample of 50 cables is tested, and it is found that the mean breaking strength is 1850 lb. Can we support the claim at 1% level of significance?
Question
b) Distinguish between Type I and Type II error. The breaking strengths of cables produced by a manufacturer have mean 1800 lb and standard deviation 100 lb. By a new technique in the manufacturing process, it is claimed that the breaking strength can be increased. To test this claim, a sample of 50 cables is tested, and it is found that the mean breaking strength is 1850 lb. Can we support the claim at 1% level of significance?
Solution
To determine if we can support the claim that the breaking strength of the cables has increased using the new manufacturing technique, we need to perform a hypothesis test.
Step 1: State the null and alternative hypotheses:
- Null hypothesis (H0): The mean breaking strength of the cables is still 1800 lb.
- Alternative hypothesis (Ha): The mean breaking strength of the cables has increased and is now greater than 1800 lb.
Step 2: Determine the significance level: In this case, the significance level is given as 1%. This means that we want to be very confident in our decision and have a low probability of making a Type I error.
Step 3: Calculate the test statistic: To calculate the test statistic, we need to use the sample mean, population mean, standard deviation, and sample size.
- Sample mean (x̄): 1850 lb
- Population mean (μ): 1800 lb
- Standard deviation (σ): 100 lb
- Sample size (n): 50
The test statistic formula for a one-sample t-test is: t = (x̄ - μ) / (σ / √n)
Plugging in the values, we get: t = (1850 - 1800) / (100 / √50)
Step 4: Determine the critical value: Since the alternative hypothesis is that the mean breaking strength has increased, we are conducting a one-tailed test. We need to find the critical value for a one-tailed test at a 1% significance level.
Using a t-table or a statistical software, we find that the critical value for a one-tailed test at a 1% significance level with 49 degrees of freedom is approximately 2.404.
Step 5: Make a decision: If the calculated test statistic is greater than the critical value, we reject the null hypothesis and support the claim. Otherwise, we fail to reject the null hypothesis.
Comparing the calculated test statistic to the critical value, if the calculated test statistic is greater than 2.404, we can support the claim that the breaking strength has increased.
Step 6: Calculate the test statistic: t = (1850 - 1800) / (100 / √50) t = 50 / (100 / √50) t = 50 / (100 / 7.071) t = 50 / 1.414 t ≈ 35.355
Step 7: Make a decision: Since the calculated test statistic (35.355) is much greater than the critical value (2.404), we can reject the null hypothesis and support the claim that the breaking strength of the cables has increased at a 1% level of significance.
Therefore, based on the given data, we can support the claim that the breaking strength of the cables has increased using the new manufacturing technique.
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