Math / Data & Statistics

QuestionAccording to the New York State Board of Law Examiners, approximately $63 \%$ of people taking the New York Bar Exam passed the exam.
1. If 20 people who have taken the New York Bar Exam are randomly selected, what is the probability that at least $76 \%$ have passed? Round your answer to 4 decimal places. $\square$
2. If 44 people who have taken the New York Bar Exam are randomly selected, what is the probability that at least $76 \%$ have passed? Round your answer to 4 decimal places. $\square$
3. Why did the probability decrease? The probability decreased since the sample size decreased resulting in a wider distribution which increased the area of the right tail. The probability decreased since the sample size increased resulting in a narrower distribution which reduced the area of the right tail.

Studdy Solution

STEP 1

What is this asking? If we grab a bunch of people who took the bar exam, what are the chances that a large percentage of them passed, given that we know the overall pass rate? Watch out! Don't mix up "at least" with "at most"!
We're looking for the probability that at least a certain percentage passed, meaning that percentage or higher!

STEP 2

1. Set up the problem
2. Calculate the first probability
3. Calculate the second probability
4. Explain the decrease

STEP 3

We're given that $\$63\%$ of people pass the bar.
So, the probability of passing (our "success") is $p = 0.63$ .

STEP 4

The problem asks for the probability that at least $76\%$ passed.
This means we're interested in the probability of $76\%$ passing, $77\%$ passing, $78\%$ passing, all the way up to $100\%$ passing.

STEP 5

We're looking at $\$20$ people. $76\%$ of $\$20$ is $0.76 \cdot 20 = 15.2$ .
Since we can't have parts of people, we need at least $\$16$ people to have passed to reach at least $\$76\%$ .

STEP 6

The binomial probability formula is perfect here!
It tells us the probability of getting exactly $k$ successes in $n$ trials: $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$ where $n$ is the number of trials (people), $k$ is the number of successes (people who passed), and $p$ is the probability of success ( $0.63$ ).

STEP 7

We need to calculate the probability of $\$16$ people passing, $\$17$ people passing, all the way up to all $\$20$ passing, and then add those probabilities together. $P(X \ge 16) = \sum_{k=16}^{20} \binom{20}{k} (0.63)^k (1-0.63)^{20-k}$ $P(X \ge 16) = \sum_{k=16}^{20} \binom{20}{k} (0.63)^k (0.37)^{20-k}$ Calculating each term and summing them: $P(X \ge 16) \approx 0.0746 + 0.0305 + 0.0088 + 0.0016 + 0.0001 \approx 0.1156$

STEP 8

Now we're looking at $\$44$ people. $76\%$ of $\$44$ is $0.76 \cdot 44 = 33.44$ .
We need at least $\$34$ people to pass.

STEP 9

Just like before, we'll use the binomial probability formula, but this time with $n = 44$ . $P(X \ge 34) = \sum_{k=34}^{44} \binom{44}{k} (0.63)^k (0.37)^{44-k}$ Calculating each term and summing them: $P(X \ge 34) \approx 0.0178 + 0.0082 + 0.0032 + 0.0010 + 0.0003 + 0.0001 + 0 + 0 + 0 + 0 + 0 \approx 0.0306$

STEP 10

As the sample size increases (from $\$20$ to $\$44$ ), the distribution gets narrower.
This means the results are more concentrated around the average.

STEP 11

Since the distribution is narrower with a larger sample size, it's less likely to see extreme results, like having a much higher pass rate than the overall average.
That's why the probability of at least $\$76\%$ passing is lower with $\$44$ people than with $\$20$ people.

STEP 12

1. For 20 people: $0.1156$
2. For 44 people: $0.0306$
3. The probability decreased since the sample size increased resulting in a narrower distribution which reduced the area of the right tail.

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Set up the problem
2. Calculate the first probability
3. Calculate the second probability
4. Explain the decrease

STEP 3

We're given that $\$63\%$ of people pass the bar.
So, the probability of passing (our "success") is $p = 0.63$ .

STEP 4

The problem asks for the probability that at least $76\%$ passed.
This means we're interested in the probability of $76\%$ passing, $77\%$ passing, $78\%$ passing, all the way up to $100\%$ passing.

STEP 5

We're looking at $\$20$ people. $76\%$ of $\$20$ is $0.76 \cdot 20 = 15.2$ .
Since we can't have parts of people, we need at least $\$16$ people to have passed to reach at least $\$76\%$ .

STEP 6

STEP 7

STEP 8

Now we're looking at $\$44$ people. $76\%$ of $\$44$ is $0.76 \cdot 44 = 33.44$ .
We need at least $\$34$ people to pass.

STEP 9

STEP 10

As the sample size increases (from $\$20$ to $\$44$ ), the distribution gets narrower.
This means the results are more concentrated around the average.

STEP 11

Since the distribution is narrower with a larger sample size, it's less likely to see extreme results, like having a much higher pass rate than the overall average.
That's why the probability of at least $\$76\%$ passing is lower with $\$44$ people than with $\$20$ people.

STEP 12

1. For 20 people: $0.1156$
2. For 44 people: $0.0306$
3. The probability decreased since the sample size increased resulting in a narrower distribution which reduced the area of the right tail.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

STEP 2

1. Set up the problem2. Calculate the first probability3. Calculate the second probability4. Explain the decrease

STEP 3

We're given that $63%\$63\%$63% of people pass the bar.So, the probability of passing (our "success") is p=0.63p = 0.63p=0.63.

STEP 4

The problem asks for the probability that *at least* 76%76\%76% passed.This means we're interested in the probability of 76%76\%76% passing, 77%77\%77% passing, 78%78\%78% passing, all the way up to 100%100\%100% passing.

STEP 5

We're looking at $20\$20$20 people. 76%76\%76% of $20\$20$20 is 0.76⋅20=15.20.76 \cdot 20 = 15.20.76⋅20=15.2.Since we can't have parts of people, we need **at least** $16\$16$16 people to have passed to reach at least $76%\$76\%$76%.

STEP 6

STEP 7

STEP 8

Now we're looking at $44\$44$44 people. 76%76\%76% of $44\$44$44 is 0.76⋅44=33.440.76 \cdot 44 = 33.440.76⋅44=33.44.We need **at least** $34\$34$34 people to pass.

STEP 9

STEP 10

As the sample size increases (from $20\$20$20 to $44\$44$44), the distribution gets narrower.This means the results are more concentrated around the average.

STEP 11

Since the distribution is narrower with a larger sample size, it's less likely to see extreme results, like having a much higher pass rate than the overall average.That's why the probability of at least $76%\$76\%$76% passing is lower with $44\$44$44 people than with $20\$20$20 people.

STEP 12

1. For 20 people: 0.11560.11560.11562. For 44 people: 0.03060.03060.03063. The probability decreased since the sample size increased resulting in a narrower distribution which reduced the area of the right tail.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Set up the problem
2. Calculate the first probability
3. Calculate the second probability
4. Explain the decrease

We're given that $\$63\%$ of people pass the bar.
So, the probability of passing (our "success") is $p = 0.63$ .

The problem asks for the probability that at least $76\%$ passed.
This means we're interested in the probability of $76\%$ passing, $77\%$ passing, $78\%$ passing, all the way up to $100\%$ passing.

We're looking at $\$20$ people. $76\%$ of $\$20$ is $0.76 \cdot 20 = 15.2$ .
Since we can't have parts of people, we need at least $\$16$ people to have passed to reach at least $\$76\%$ .

Now we're looking at $\$44$ people. $76\%$ of $\$44$ is $0.76 \cdot 44 = 33.44$ .
We need at least $\$34$ people to pass.

As the sample size increases (from $\$20$ to $\$44$ ), the distribution gets narrower.
This means the results are more concentrated around the average.

Since the distribution is narrower with a larger sample size, it's less likely to see extreme results, like having a much higher pass rate than the overall average.
That's why the probability of at least $\$76\%$ passing is lower with $\$44$ people than with $\$20$ people.

1. For 20 people: $0.1156$
2. For 44 people: $0.0306$
3. The probability decreased since the sample size increased resulting in a narrower distribution which reduced the area of the right tail.