Final answer:
To determine the level of confidence with which an engineer's claim can be assessed, statistical hypothesis testing, specifically the z-test, is used based on sample means, standard deviations, and sample sizes. A z-score is calculated to find the p-value, which is then compared against a significance level to test the claim. Larger sample sizes increase the confidence level of the statistical test results.
Step-by-step explanation:
To assess the claim about the mean lifetime of batteries, we need to apply concepts of statistical hypothesis testing. With a sample mean life span of 16.7 hours for 30 batteries when the claimed average is 17 hours, and a standard deviation of 0.8 hours, we can use the z-test to calculate the probability (p-value) of observing such a sample mean if the true mean is as claimed. Similarly, to test the tire lifespan and work week claims, we use statistical tests with the given sample means, standard deviations, and sample sizes to determine if the observed samples are significantly different from the claims. These comparisons produce p-values which, when compared against a chosen significance level (typically 0.05), indicate whether the claims about the mean can be reasonably doubted.
For example, the z-score for the batteries can be found using the formula for the standard error of the mean SEM = standard deviation / √n (where n is the sample size), and then finding the z-score by subtracting the sample mean from the claimed mean and dividing by SEM. After calculating the z-score, we would use a z-table (or normal distribution function in a calculator) to find the corresponding p-value. If the p-value is less than the predetermined significance level alpha (e.g., 0.05), then we have evidence to doubt the claim.
A larger sample size tends to produce a more accurate estimate and can lead to a higher level of confidence in the result. This means that, for the Swedish survey example, surveying 1,000 male Swedes instead of 48 would generally increase the confidence in the results. This is because larger samples tend to better represent the population and reduce the margin of error.