A manufacturer of nickel-hydrogen batteries randomly selects 100 nickel plates for test cells, cycles them a specified number of times, and determines that 14 of the plates have blistered. (a) Does this provide compelling evidence for concluding that more than 10% of all plates blister under such circumstances? State and test the appropriate hypotheses using a significance level of .05. In reaching your conclusion, what type of error might you have committed? (b) If it is really the case that 15% of all plates blister under these circumstances and a sample size 100 is used, how likely is it that the null hypothesis of part (a) will not be rejected by the 0.05 test? Answer this question for a sample size of 200. (c) How many plates would have to be tested to have \(\beta(.15)\) = 0.10 for the test of part (a)?

(a) To determine if more than 10% of all nickel plates blister under the specified conditions, we can set up a hypothesis test.

Null Hypothesis (H0): p = 0.10 (10% of the plates blister) Alternative Hypothesis (H1): p > 0.10 (More than 10% of the plates blister)

Where p is the true blister rate.

We calculate the test statistic using the formula for the proportion test:

Test Statistic (z) = (p-hat - p0) / sqrt[(p0(1-p0))/n]

Where: - p-hat is the sample proportion (14/100 in this case) - p0 is the hypothesized population proportion (0.10 in this case) - n is the sample size (100)

p-hat = 14/100 = 0.14

z = (0.14 - 0.10) / sqrt[(0.10 * 0.90) / 100] z = 0.04 / sqrt[0.09 / 100] z = 0.04 / 0.03 z ≈ 1.33

Now we check this z-value against the normal distribution for a one-tailed test at the significance level α = 0.05. The critical z-value for α = 0.05 is approximately 1.645.

Since our calculated z-value (1.33) is less than the critical z-value (1.645), we fail to reject the null hypothesis. This means we do not have compelling evidence to conclude that more than 10% of all plates blister.

Type of error:

If in reality more than 10% of the plates do blister but our test has failed to detect this, we may have committed a Type II error (failing to reject a false null hypothesis).

(b) To find the probability that the null hypothesis will not be rejected when in fact 15% of all plates blister, we need to calculate the probability of a Type II error (β).

For a sample size of 100: The power of the test is the probability of correctly rejecting a false null hypothesis, which is 1 - β.

p = 0.15 (actual blister rate) n = 100 Using the standard normal distribution, we can find the z-value where the null hypothesis will not be rejected.

Critical z-value for α = 0.05 (one-tailed) is approximately 1.645.

To calculate z, we use p = 0.15 and figure out where it lies relative to p0 = 0.10:

z = (p - p0) / sqrt[(p0(1-p0))/n] z = (0.15 - 0.10) / sqrt[(0.10 * 0.90) / 100] z = 0.05 / 0.03 z ≈ 1.67

We find the corresponding β by looking up the value of 1.67 in the z-tables. This gives us a probability (β) that the null hypothesis will not be rejected when the true proportion is 0.15.

For a sample size of 200, the standard error would decrease, leading to a higher z-value and potentially a lower β (increased power of the test). Repeat the same calculation with n = 200 to determine the new β for that sample size.

(c) To find the required sample size to have β(.15) = 0.10 for the test of part (a), we need to work backwards using the power calculation. We know that when p0 = 0.10 and the actual proportion p = 0.15, we want the power of the test (1 - β) to be 0.90 (so β = 0.10).

Using the standard normal distribution and working backwards, we can determine the sample size that gives us a test statistic corresponding to a β of 0.10 when the critical value for α = 0.05. This requires iterative calculation or specific statistical software to arrive at the precise sample size that satisfies this condition. In practice, however, sample size calculators are available online or within statistical packages that allow one to input the desired power, significance level, and effect size to find the necessary sample size.

Extra: The concepts discussed here are fundamental to hypothesis testing in statistics:

- Null Hypothesis (H0) is the default statement that no effect or no difference is present. - Alternative Hypothesis (H1) represents what the researcher wants to prove or establish. - Significance Level (α) is the probability of rejecting the null hypothesis when it is actually true, often set at 0.05 or 5%. - Type I Error is incorrectly rejecting a true null hypothesis (also known as a "false positive"). - Type II Error is failing to reject a false null hypothesis (also known as a "false negative"). - Test Statistic is a standardized value that results from a statistical test calculation. - Power of a Test is the probability that the test will correctly reject a false null hypothesis. - β (beta) is the probability of a Type II error, and (1 - β) is the power. - Sample Size (n) plays a critical role in hypothesis testing; a larger sample size generally increases the test's power.

Understanding these concepts can help students make informed decisions when designing experiments and interpreting statistical results.