Question

question 6
a scientist is rolling a die which has six possible outcomes. he is comparing the expected values of each number to see if the die is equally weighted. the chi - squared value of his experiment is 7.84. using the 95% confidence interval (0.05 probability value) what is the degrees of freedom for this experiment?
a 6
b 5
c 7
d 4
question 7
10 points
if we flipped a coin 100 times and we got heads 59 times would this to be significantly different from the 50:50 ratio?
a yes
b no

Explanation:

Step1: Recall degrees - of - freedom formula

For a chi - squared goodness - of - fit test with $k$ categories, the degrees of freedom is $df=k - 1$.

Step2: Identify number of categories

When rolling a die, there are $k = 6$ possible outcomes (the numbers 1 - 6).

Step3: Calculate degrees of freedom

$df=6 - 1=5$.

For the coin - flipping problem:

Step1: Calculate expected and observed values

The expected number of heads in 100 flips of a fair coin is $E = 100\times0.5=50$, and the observed number of heads is $O = 59$.

Step2: Calculate chi - squared statistic

The chi - squared statistic formula is $\chi^{2}=\sum\frac{(O - E)^{2}}{E}$. Here, $\chi^{2}=\frac{(59 - 50)^{2}}{50}+\frac{(100 - 59-50)^{2}}{50}=\frac{9^{2}}{50}+\frac{(- 9)^{2}}{50}=\frac{81 + 81}{50}=\frac{162}{50}=3.24$.

Step3: Determine degrees of freedom

For a coin - flip (2 categories: heads and tails), $df=2 - 1 = 1$.

Step4: Compare with critical value

For $df = 1$ and a 95% confidence interval ($\alpha=0.05$), the critical chi - squared value is 3.841. Since $3.24<3.841$, the result is not significantly different from the 50:50 ratio.

Answer:

Question 6: B. 5
Question 7: B. No