Question

For which of the following \(P\) -values will the null hypothesis be rejected when performing a test with a significance level of .05: a. .001 d. .047 b. .021 e. .148 c. .078

Short answer

For \(P\) values of .001, .047, and .021, the null hypothesis would be rejected when performing a test with a significance level of .05.

Step-by-step solution

Compare the first \(P\) value with the significance level

The first provided \(P\) value is .001. As .001 is less than the significance level of .05, for this value of \(P\), the null hypothesis would be rejected.

Compare the second \(P\) value with the significance level

The second provided \(P\) value is .047. As .047 is less than the significance level of .05, for this value of \(P\), the null hypothesis would also be rejected.

Compare the third \(P\) value with the significance level

The third provided \(P\) value is .021. As .021 is also less than the significance level of .05, for this value of \(P\), again, the null hypothesis would be rejected.

Compare the fourth \(P\) value with the significance level

The fourth provided \(P\) value is .148. But this time, .148 is greater than the significance level of .05. So, for this value of \(P\), the null hypothesis would not be rejected.

Compare the fifth \(P\) value with the significance level

The fifth and last provided \(P\) value is .078. As .078 is greater than the significance level of .05, for this value of \(P\), the null hypothesis would not be rejected either. Thus, resulting in a total of three \(P\) values (.001, .047, .021) where the null hypothesis would be rejected for a significance level of .05.

Key concepts

P-value

The P-value is a probability that helps determine the strength of the results from a hypothesis test. Essentially, it tells us how likely it is to observe the test results under the assumption that the null hypothesis is true. When analyzing P-values, the smaller the number, the stronger the evidence against the null hypothesis.

• If the P-value is small (typically less than the chosen significance level), it suggests that the observed data is unlikely under the null hypothesis.
• Conversely, a large P-value indicates that the observed data is not surprising under the null hypothesis.

It's like flipping a coin—if you expect it to be fair (50% heads and 50% tails) but see it land on heads every time, a low P-value would suggest the coin could be biased.

null hypothesis

In hypothesis testing, the null hypothesis ( \(H_0\)) is a statement that indicates no effect or no difference. It is the default position that there is nothing unusual happening.

• The null hypothesis acts as a starting point for statistical testing.
• It's said to be 'accepted' if the test results show insufficient evidence against it, meaning the P-value is higher than the significance level (alpha).

For instance, if we are testing a new drug, the null hypothesis would state that the drug has no effect compared to the existing treatment. The main goal is to use sample data to determine if this hypothesis can be rejected.

significance level (alpha)

The significance level, often denoted by alpha ( \(\alpha\)), is a threshold set by the researcher which defines how much evidence is required to reject the null hypothesis.

• A common significance level is 0.05, but it can be adjusted based on the researcher's criteria and the context of the test.
• If the P-value is less than or equal to the significance level, it suggests that the evidence is strong enough to reject the null hypothesis.

Choosing an appropriate alpha level involves balancing the risk of making a Type I error (rejecting the null hypothesis when it is true) against the need to detect a true effect. Setting a lower alpha level means more evidence is needed to reject the null hypothesis, reducing the risk of falsely detecting an effect.