5 min read•june 18, 2024
Josh Argo
Jed Quiaoit
Josh Argo
Jed Quiaoit
The of a is a measure of the probability of obtaining a sample with a test statistic that is at least as extreme as the one observed, under the assumption that the is true. In other words, it's the proportion of possible samples of a given size that are equal to or less than/greater than our given sample. 🧺
It is used to help determine whether the observed results are or not.
The p-value is the "proportion of values for the null distribution that are as extreme or more extreme than the observed value of the test statistic." This is
If our p-value is low, this means that it is highly unlikely that our sample would be chosen randomly. This could be due to one of three things: 🤷
Again, it's important to remember that the p-value is computed under the assumption that the null hypothesis is true. Therefore, when interpreting the p-value of a significance test, it's important to consider the context in which the test was conducted and the implications of the null hypothesis being true! ⚠️
For example, in a , the null hypothesis typically states that the true population proportion is equal to a particular value (usually 0.5, or no difference from the hypothesized value). If the p-value is small, it suggests that the observed sample proportion is significantly different from the hypothesized value, and provides evidence against the null hypothesis. In this case, you might conclude that the true population proportion is different from the hypothesized value.
On the other hand, if the p-value is large, it suggests that the observed sample proportion is not significantly different from the hypothesized value, and does not provide strong evidence against the null hypothesis. In this case, you might conclude that there is insufficient evidence to reject the null hypothesis and that the true population proportion is equal to the hypothesized value.
In the recent issue of Sports Unlimited, Jackie reads that a right-handed hockey player scores on approximately 5% of their shots. To test this claim, Jackie watches 15 random hockey games and records 921 shots from random, right-handed hockey players. She finds that they scored on 60 of those shots. After calculating her z-score and p-value, she finds that her p-value is essentially 0.017. Interpret this p value. 🏒
This p-value means that of all possible samples of 921 shots from right handed players, approximately 1.7% of those samples would have at least 60 shots. This sample was random from the given information, so no obvious sampling bias. It could be that Jackie just hit the jackpot and watched the right players to have such a high goal scoring percentage. She could check this by redoing the experiment a few times.
The other option is that the 5% isn't actually correct. Maybe that hypothesized percentage is a bit higher... 🤔
🎥 Watch: AP Stats - Inference: Hypothesis Tests for Proportions
A political campaign is trying to determine whether the proportion of registered voters in their district who support their candidate is significantly different from the overall national proportion of 50%. They conduct a survey by randomly sampling 1000 registered voters in their district and ask whether they support their candidate. They find that 540 out of the 1000 respondents support their candidate. 📋
a) Write the for this scenario.
b) After conducting a one-sample z-test to determine whether the proportion of registered voters in the district who support the candidate is significantly different from the national proportion of 50%, you find that the p-value for this one-sample z-test is 0.031. Based on the results of the z-test and the p-value, what can the campaign conclude about the proportion of registered voters in the district who support their candidate? What are the limitations of this conclusion?
a) Null hypothesis: The proportion of registered voters in the district who support the candidate is equal to the national proportion of 50%.
H0: p = 0.50
Alternative hypothesis: The proportion of registered voters in the district who support the candidate is significantly different from the national proportion of 50%.
Ha: p ≠ 0.50
b) Based on the results of the z-test, the campaign can conclude that the proportion of registered voters in the district who support their candidate is significantly different from the national proportion of 50%, because the p-value of 0.031 is smaller than the commonly used significance level of 0.05.
This suggests that the proportion of registered voters in the district who support the candidate is higher than the national proportion. However, it's important to note that this conclusion is based on the assumption that the null hypothesis (that the proportion of registered voters in the district who support the candidate is equal to the national proportion of 50%) is true.