Average Lifetime Hypothesis Testing
A researcher is interested in finding out whether the average lifetime of females in the US is different from 75 years. For this, he takes a sample of 100 females with a sample mean of 76 and a sample standard deviation of 7. State the null and alternative hypotheses at a 95% confidence level. Is there enough evidence to reject the null hypothesis?
In this problem, you will explore the concept of hypothesis testing, a fundamental aspect of inferential statistics used to determine if there is enough evidence to reject a presumed assumption about a population parameter. Specifically, you are tasked with assessing whether the average lifetime of females in the US differs from a specified value, using sample data.
The null hypothesis is a statement of no effect or no difference, often representing a status quo belief. Here, it would state that the average lifetime of females in the US is equal to 75 years. The alternative hypothesis, on the other hand, contradicts the null and suggests that the population mean lifetime is different from 75 years. This problem requires setting these hypotheses clearly as the groundwork for the test and making a decision based on statistical evidence.
Understanding how to conduct a hypothesis test involves familiarity with concepts such as sample means, confidence levels, and standard deviations. You must explore whether the observed sample mean of 76 years provides enough statistical evidence to cast doubt on the null hypothesis by considering the variability captured by the sample standard deviation. This test is typically conducted using a t-distribution when the population standard deviation is unknown, but since your sample size is large, the normal approximation applies.
Deciding whether there is enough evidence to reject the null hypothesis involves comparing a calculated test statistic to a critical value from the significance level, determined by the confidence level. A 95% confidence level implies a 5% significance level, setting a specific threshold for making this decision. An understanding of these probabilistic thresholds and their implications is crucial when interpreting results, ensuring well-founded conclusions from the data analysis.
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