Suppose the quality control manager for a cereal manufacturer wants to ensure that bags of cereal are filled correctly. The equipment is set to fill bags with a mean of 16 oz of cereal with a standard deviation of 0.2 oz. The quality control inspector selects a random sample of 45 boxes and finds that the mean amount of cereal in these boxes is 16.03 oz. He uses this data to conduct a hypothesis test: H0: μ = 16, where μ is the mean amount of cereal in each box. He calculates a z-score of 1.01 and a p-value of 0.3125. Are these results statistically significant at a significance level of 0.05? O No, these results are not statistically significant because p > 0.05. O No, these results are not statistically significant because p < 0.05. O Yes, these results are statistically significant because p < 0.05. O Yes, these results are statistically significant because p > 0.05. The correct option is: O No, these results are not statistically significant because p > 0.05.

The correct option is:

O No, these results are not statistically significant because p > 0.05.

In hypothesis testing, the p-value is a measure of the probability of obtaining a result at least as extreme as the one observed in the sample data, under the assumption that the null hypothesis (H0) is true. If the p-value is less than the chosen significance level (α), which is often set at 0.05, then the result is considered statistically significant. This means that it's unlikely the observed data would occur by random chance if the null hypothesis were true, and therefore the null hypothesis is rejected.

In this scenario, the p-value calculated from the test is 0.3125. Since this p-value is greater than the significance level of 0.05, we do not have sufficient evidence to reject the null hypothesis. Therefore, the results are not statistically significant, and we would conclude that the data does not provide strong evidence against the null hypothesis that μ = 16. The difference between the sample mean and the hypothesized population mean (16.03 oz vs. 16 oz) could easily be due to random sampling variability.

Extra: Statistical significance is a fundamental concept in hypothesis testing. A significance level of 0.05 is a common threshold, meaning there is a 5% chance of rejecting the null hypothesis when it is actually true (Type I error). The z-score is another important metric in hypothesis testing when dealing with normal distributions. The z-score represents the number of standard deviations a data point is from the mean. A z-score of 1.01 means that the sample mean is 1.01 standard deviations above the hypothesized population mean.

The choice of the significance level depends on the context of the study and how much risk of a Type I error the researchers are willing to accept. In highly sensitive areas of research, a lower significance level, such as 0.01, might be used to reduce the chance of Type I error.

When we fail to reject the null hypothesis, as is the case with the cereal manufacturer study, it doesn't necessarily mean that the null hypothesis is true. It simply means that there's not enough evidence to conclude that it is false, based on the data and the significance level chosen. The manufacturer's quality control process as measured in this sample appears to be performing within the specified parameters, and there's insufficient evidence to suggest a deviation from the expected mean fill of 16 oz.