Question

1. if bianca rolled 22 sixes, then what would be the most likely amount of times she rolled her die? 2. if someone rolled their die 30 times and got 18 sixes, would you consider that to be an outlier? 3. if peter rolled a die 180 times, how many sixes would you expect him to get?

Explanation:

Step1: Analyze the scatter - plot for the first question

By observing the scatter - plot, when the number of sixes is 22, the most likely number of times the die is rolled is around 130.

Step2: Calculate expected value for the second question

The probability of getting a six on a fair die roll is $p=\frac{1}{6}$. When rolling a die $n = 30$ times, the expected number of sixes is $E(X)=np=30\times\frac{1}{6}=5$. The standard deviation for a binomial distribution is $\sigma=\sqrt{np(1 - p)}=\sqrt{30\times\frac{1}{6}\times(1-\frac{1}{6})}=\sqrt{30\times\frac{1}{6}\times\frac{5}{6}}=\sqrt{\frac{25}{6}}\approx2.04$. Using the range - rule for significant values (usual values are within $E(X)\pm2\sigma$), the lower limit is $5-2\times2.04 = 0.92$ and the upper limit is $5 + 2\times2.04=9.08$. Since 18 is well outside this range, it is an outlier.

Step3: Calculate expected value for the third question

For a fair die, the probability of getting a six $p=\frac{1}{6}$. When $n = 180$, the expected number of sixes is $E(X)=np=180\times\frac{1}{6}=30$.

Answer:

1. 130
2. Yes
3. 30