A special industrial battery must have a life of at least...

A special industrial battery must have a life of at least 400 hours. A hypothesis test is to be conducted with a .02 level of significance. If the batteries from a particular production run have an actual mean use life of 385 hours, the production manager wants a sampling procedure that only $10 \%$ of the time would show erroneously that the batch is acceptable. What sample size is recommended for the hypothesis test? Use 30 hours as an estimate of the population standard deviation.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical inference procedure that allows us to evaluate assumptions about a population parameter. In this context, it involves setting up a null hypothesis (typically representing the status quo or acceptable condition) and an alternative hypothesis (indicating a departure from that condition) and then using sample data to decide whether to reject the null hypothesis.

Significance Level

The significance level (denoted by ?) is the probability of making a Type I error, which is rejecting a true null hypothesis. It defines the threshold for the test statistic below which the null hypothesis will be rejected. Here, an ? of 0.02 means there is a 2% risk of erroneously concluding that a battery batch meets the required life span when it does not.

Type II Error and Power of a Test

A Type II error occurs when a test fails to reject a false null hypothesis. The probability of a Type II error is denoted by ?, while the power of the test (1 ? ?) reflects the test’s ability to correctly reject a false null hypothesis. In this scenario, the production manager desires the test procedure to erroneously conclude acceptance (a Type II error) only 10% of the time when the actual mean life is below the acceptable threshold.

Sample Size Determination

Sample size determination involves calculating the number of observations required to achieve a desired level of test sensitivity and error control. A larger sample size reduces the standard error of the mean, thereby increasing the test’s power (its ability to detect a true effect) and reducing the likelihood of both Type I and Type II errors.

Effect Size

Effect size in this context refers to the magnitude of the difference between the hypothesized minimum acceptable mean (e.g., 400 hours) and the actual mean (e.g., 385 hours). The effect size, combined with the variability in the population (as measured by the standard deviation), influences how large a sample is needed to reliably detect the difference.

Z-Test for Means

A Z-test for means is used when the population standard deviation is known and the sample size is sufficiently large. This test assesses whether the sample mean is significantly different from the hypothesized population mean. It employs standardization, converting the sample mean to a Z-score using the known standard deviation, which can then be compared against critical values from the standard normal distribution.

Transcript

Please give Ace some feedback