Question

1. mrs. g is very good at sudoku puzzles. 75% of the time, she can complete a puzzle in under 10 minutes. her class challenges her to test this success rate by completing 10 puzzles.

a) define a random variable x and show that it follows a binomial distribution.
x = # of complete a puzzle under 10 min.
two outcomes: complete / not complete
fixed probability: p = .75
independent trials: ✓
fixed trials: 10
b) complete the probability distribution below.

x	p(x)
0	0
1	0.000029
2	0.000386
3	0.00309
4	0.016222
5	0.058399
6	0.145998
7	0.250282
8	0.281568
9	0.187712
10	0.056314

c) find and interpret the expected value.
\\(\mu_x = np\\)
\\(= 10 \times .75\\)
\\(= 7.5\\)
mrs. g expect to success 7.5 times out of 10
d) find and interpret the standard deviation.
\\(\sigma = \sqrt{np(1 - p)} = \sqrt{10 \times .75(1 - .75)}
=\\)
e) if mrs. g completes all 10 puzzles in under 10 minutes each, would you be suspicious of her cheating? explain why or why not using probability.

1. a certain midwestern university claims that 90% of their football players complete their degree in 5 years. the ncaa investigates this claim by selecting a random sample of 20 football players that have been part of the program within the past 5 years. 14 of these players received their degree, while 6 did not receive their degree. if you were the investigator, what would you conclude about the university’s claim? explain your reasoning with probability.

Explanation:

Step1: Calculate standard deviation

For a binomial distribution, the formula for standard deviation is $\sigma=\sqrt{np(1 - p)}$, where $n = 10$ (number of trials) and $p=0.75$ (probability of success).
$$\sigma=\sqrt{10\times0.75\times(1 - 0.75)}$$

Step2: Simplify the expression

First, calculate $10\times0.75\times(1 - 0.75)=10\times0.75\times0.25 = 1.875$. Then, find the square - root of $1.875$.
$$\sigma=\sqrt{1.875}\approx1.37$$

Step3: Interpret the standard deviation

The standard deviation $\sigma\approx1.37$ means that, on average, the number of puzzles Mrs. G completes in under 10 minutes in a set of 10 puzzles deviates from the mean value of $\mu = 7.5$ (from part c) by about $1.37$ puzzles.

Step4: Analyze part e

The probability that $X = 10$ (completing all 10 puzzles in under 10 minutes) is $P(X = 10)=0.056314$. A probability of approximately $5.63\%$ is relatively low. While it is not extremely rare (a common cut - off for “rare” is often $5\%$ or lower in some contexts, but it's a bit of a grey area), it is on the lower side of what we would expect based on her stated success rate of $75\%$. So, it might be a cause for some suspicion, but it's not an extremely conclusive result.

Answer:

• For part d: The standard deviation $\sigma\approx1.37$. It means the number of puzzles Mrs. G completes in under 10 minutes in a set of 10 puzzles deviates from the mean of $7.5$ by about $1.37$ puzzles.
• For part e: The probability of completing all 10 puzzles ($P(X = 10)=0.056314$) is relatively low (about $5.63\%$). While not extremely rare, it could potentially raise some suspicion about her cheating as it is on the lower end of what we might expect from a $75\%$ success rate.