Problem 1 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

We perform a \(t\)-test for the null hypothesis \(H_{0}: \mu=10\) by means of a dataset consisting of \(n=16\) elements with sample mean 11 and sample variance 4 . We use significance level \(0.05\). a. Should we reject the null hypothesis in favor of \(H_{1}: \mu \neq 10\) ? b. What if we test against \(H_{1}: \mu>10 ?\)

Step-by-Step Solution

Answer

(a) Do not reject; (b) Reject the null hypothesis.

Step 1: Identify the test statistic formula

To conduct a hypothesis test, we first need to calculate the test statistic using the formula for a one-sample t-test: \( t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \). In this case, \( \bar{x} = 11 \), \( \mu = 10 \), \( s^2 = 4 \) (thus \( s = 2 \)), and \( n = 16 \).

Step 2: Calculate the test statistic

Using the formula from Step 1: \( t = \frac{11 - 10}{2/\sqrt{16}} = \frac{1}{0.5} = 2 \). The calculated t-value is 2.

Step 3: Determine critical values

For a two-tailed test at a significance level of \( \alpha = 0.05 \), with \( n - 1 = 15 \) degrees of freedom, refer to a t-distribution table to find the critical values. The critical value is approximately \( \pm 2.131 \).

Step 4: Decision for part (a)

The calculated test statistic \( t = 2 \) falls within the critical region \((-2.131, 2.131)\). Thus, we do not have sufficient evidence to reject the null hypothesis for part (a).

Step 5: Determine critical value for one-tailed test

For a one-tailed test at \( \alpha = 0.05 \) and 15 degrees of freedom, the critical t-value is approximately \( 1.753 \).

Step 6: Decision for part (b)

The calculated t-value \( t = 2 \) is greater than the critical value \( 1.753 \) for the one-tailed test. Thus, we reject the null hypothesis in favor of \( H_1: \mu > 10 \).

Key Concepts

Null Hypothesis

In statistics, when we conduct a hypothesis test, we start by setting up a null hypothesis (denoted as \(H_0\)). The null hypothesis is a statement we initially assume to be true. It typically suggests there is no effect or no difference. For example, in our exercise, the null hypothesis is \(H_0: \mu = 10\). This means we claim the population mean \(\mu\) is 10. Our goal is to test this claim with our sample data. The null hypothesis acts as a starting point in our statistical test. It's like assuming innocence in a trial until proven guilty. We gather evidence to see if we can reject \(H_0\) or not.

Significance Level

The significance level, symbolized by \(\alpha\), is an important concept in hypothesis testing. It helps us determine the threshold for decision making. The significance level is the probability of rejecting the null hypothesis when it's true. In simpler terms, it's the risk we're willing to take to say there's an effect when there actually isn't one.

In the exercise, we use a significance level of \(0.05\). This means we accept a 5% chance of making a wrong decision by rejecting \(H_0\) incorrectly. Typically, \(\alpha\) is set at common values like 0.05, 0.01, or 0.10, depending on the context of the problem and the level of confidence required. Setting \(\alpha\) helps in comparing the calculated test statistic against the critical values to make a decision.

T-Distribution

The \(t\)-distribution is a crucial concept when dealing with small sample sizes, especially when the population standard deviation is unknown. It closely resembles the standard normal distribution but with heavier tails. This means more probability in the tails for small sample sizes, accommodating more variation.

The \(t\)-distribution depends on degrees of freedom, typically calculated as \(n-1\) where \(n\) is the sample size. In our exercise, with \(n = 16\), the degrees of freedom are 15. As the sample size increases, the \(t\)-distribution approaches the normal distribution. We refer to the \(t\)-distribution table to find critical values when conducting our \(t\)-test, adjusting for the degrees of freedom and significance level.

Critical Values

Critical values are threshold points that help determine whether the calculated test statistic leads us to reject the null hypothesis. These values are derived from the \(t\)-distribution table based on the significance level (\(\alpha\)) and degrees of freedom.

• For a two-tailed test, we split \(\alpha\) into two, finding critical values on both ends of the distribution. In our exercise, the critical values are approximately \(\pm 2.131\) for \(\alpha = 0.05\) and 15 degrees of freedom.
• For a one-tailed test, we only consider one side of the distribution, depending on the direction of the alternative hypothesis. The critical value is \(1.753\) for a one-tail test with the same settings.

These values guide us in decision-making: if the test statistic falls beyond these critical values, we reject the null hypothesis; if not, we fail to reject it, meaning the evidence is not strong enough to support the alternative hypothesis.