Question

A confidence interval for a sample is given, followed by several hypotheses to test using that sample. In each case, use the confidence interval to give a conclusion of the test (if possible) and also state the significance level you are using. A \(99 \%\) confidence interval for \(\mu: 134\) to 161 (a) \(H_{0}: \mu=100\) vs \(H_{a}: \mu \neq 100\) (b) \(H_{0}: \mu=150 \mathrm{vs} H_{a}: \mu \neq 150\) (c) \(H_{0}: \mu=200\) vs \(H_{a}: \mu \neq 200\)

Short answer

For part (a), the null hypothesis \(H_{0}: \mu=100\) can be rejected. For part (b), there's not enough evidence to reject the null hypothesis \(H_{0}: \mu=150\). Lastly, for part (c), the null hypothesis \(H_{0}: \mu=200\) can be rejected.

Step-by-step solution

Understanding Confidence Interval

A 99% confidence interval is a type of estimate computed from the statistics of the observed data that might contain the true value of an unknown population parameter. The interval has an associated confidence level that quantifies the level of confidence that the parameter lies within the interval. In this case, the 99% confidence interval for the population parameter \(\mu\) is between 134 and 161.

Testing Hypotheses (a)

The null hypothesis for part (a) is \(H_{0}: \mu=100\). With the given 99% confidence interval for \(\mu: 134\) to 161, it can be observed that 100 is not within this interval. Therefore, this null hypothesis can be rejected with 99% confidence.

Testing Hypotheses (b)

In part (b), the null hypothesis is \(H_{0}: \mu=150\). Given the confidence interval of 134 to 161, we can see that 150 falls within this interval. Therefore, there's not enough evidence to reject the null hypothesis at the 99% confidence level.

Testing Hypotheses (c)

For part (c), the null hypothesis is \(H_{0}: \mu=200\). Checking this against the confidence interval (134 to 161), it's clear that 200 does not fall within this interval. This implies that the null hypothesis can hence be rejected with 99% confidence.

Key concepts

Hypothesis Testing

Hypothesis testing is a statistical procedure used to make decisions about a population parameter based on sample data. It is essentially a method for determining the validity of a conjecture (also called hypothesis) about a population.

The process starts by proposing two hypotheses: the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)). The null hypothesis usually states that there is no effect or no difference, and it sets the stage for statistical significance testing. One conducts an experiment or derives a sample to collect data, and uses statistical tests to determine whether the evidence is strong enough to reject the null hypothesis in favor of the alternative.

For instance, if a sample suggests a significantly different outcome than what the null hypothesis predicts, you might have reason to reject it, leading to a conclusion in favor of the alternative hypothesis. In educational contexts, understanding the steps involved in this process and the logic behind hypothesis testing can turn seemingly abstract concepts into concrete analysis tools.

Statistical Significance

Statistical significance refers to the likelihood that a result from data sampled from a population is not due to random chance. It plays a critical role in hypothesis testing as it helps us to determine if the evidence is strong enough to make a reliable decision about rejecting the null hypothesis.

When a result is statistically significant, it means the observations have a very low probability of occurring if the null hypothesis were true (usually below a predefined threshold called the significance level, e.g., 5%, 1%). In our exercise, a 99% confidence interval implies a 1% significance level, meaning that there is less than a 1% chance that the sample results are due to random variation, assuming the null hypothesis is true.

Understanding how statistical significance impacts hypothesis testing empowers students to comprehend why certain conclusions are drawn from data and the strength of the evidence supporting those conclusions. It's crucial to remember that statistical significance does not speak to the practical importance of a finding, only the confidence in its non-randomness.

Null Hypothesis

The null hypothesis (\(H_0\)) is the default assumption that there is no difference or effect regarding the population parameter in question. It serves as the starting point for statistical significance testing.

In hypothesis testing, we aim to determine whether enough evidence is present to reject the null hypothesis. The burden of proof lies with the alternative hypothesis, which is a statement that contradicts the null hypothesis, positing that there is indeed an effect or a difference.

Interpreting the Null Hypothesis with Confidence Intervals

In the context of confidence intervals, if the interval includes the value specified in the null hypothesis, we do not have enough statistical evidence to reject it. For example, in our original exercise, since the 99% confidence interval for \(\mu\) includes 150, we would not reject the null hypothesis \(H_0: \mu=150\). Conversely, values outside the interval, such as 100 and 200 in the examples given, would lead us to reject the null hypothesis. This concept is crucial for students to understand as it forms the basis for determining the plausibility of statistical claims and making decisions in the face of uncertainties inherent in sample data.