Research question Section
When randomly selecting a card from a deck with replacement, are we equally likely to select a heart, diamond, spade, and club?
I randomly selected a card from a standard deck 40 times with replacement. I pulled 13 hearts, 8 diamonds, 8 spades, and 11 clubs.
Let's use the five-step hypothesis testing procedure:
\(H_0: p_h=p_d=p_s=p_c=0.25\)
\(H_a:\) at least one \(p_i\) is not as specified in the null
We can use the null hypothesis to check the assumption that all expected counts are at least 5.
\(Expected\;count=n (p_i)\)
All \(p_i\) are 0.25. \(40(0.25)=10\), thus this assumption is met and we can approximate the sampling distribution using the chi-square distribution.
\(\chi^2=\sum \dfrac{(Observed-Expected)^2}{Expected} \)
All expected values are 10. Our observed values were 13, 8, 8, and 11.
\(\chi^2=\dfrac{(13-10)^2}{10}+\dfrac{(8-10)^2}{10}+\dfrac{(8-10)^2}{10}+\dfrac{(11-10)^2}{10}\)
\(\chi^2=\dfrac{9}{10}+\dfrac{4}{10}+\dfrac{4}{10}+\dfrac{1}{10}\)
\(\chi^2=1.8\)
Our sampling distribution will be a chi-square distribution.
\(df=k-1=4-1=3\)
We can find the p-value by constructing a chi-square distribution with 3 degrees of freedom to find the area to the right of \(\chi^2=1.8\)
The p-value is 0.614935
\(p>0.05\) therefore we fail to reject the null hypothesis.
There is not evidence that the proportion of hearts, diamonds, spades, and clubs that are randomly drawn from this deck are different.