Math / Data & Statistics

QuestionFrom historical data, $18 \%$ of people work out daily $(p=0.18)$ . In a sample of 15 $(n=15)$ people, use the binomial formula or your TI-8314 to determine the probability that exactly 4 (of the fifteen) work out daily.
For a standard normal distribution: a) What is the probability that $z$ is less than 0.47 ? b) What is the probability that $z$ is greater than -1.83 ? $\qquad$ c) What is the probability that $z$ is between -1.29 and 0.64 ? $\qquad$ d) What is the probability that $z$ is between -1.96 and 1.96 ? $\qquad$

Studdy Solution

STEP 1

What is this asking? Out of 15 people, what's the chance that exactly 4 of them work out daily, if we know 18% of people generally do?
Also, we need to find some probabilities using a z-table or calculator. Watch out! Don't mix up the binomial formula parts, and make sure to use the correct z-scores for the normal distribution questions!

STEP 2

1. Binomial Probability
2. Z-score Less Than
3. Z-score Greater Than
4. Z-score Between Two Values (1)
5. Z-score Between Two Values (2)

STEP 3

The binomial probability formula tells us the chance of getting exactly $k$ successes in $n$ trials: $P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{(n-k)}$ Where $p$ is the probability of success on a single trial.

STEP 4

We have $n = \textbf{15}$ people (trials), we want $k = \textbf{4}$ people to work out (successes), and the probability of someone working out daily is $p = \textbf{0.18}$ .
Let's plug these into our formula! $P(X=4) = \binom{15}{4} \cdot (0.18)^4 \cdot (1-0.18)^{(15-4)}$

STEP 5

The binomial coefficient $\binom{15}{4}$ represents the number of ways to choose 4 people out of 15.
It's calculated as: $\binom{15}{4} = \frac{15!}{4! \cdot (15-4)!} = \frac{15!}{4! \cdot 11!} = \frac{15 \cdot 14 \cdot 13 \cdot 12}{4 \cdot 3 \cdot 2 \cdot 1} = \textbf{1365}$

STEP 6

Now, let's plug everything back into our binomial formula: $P(X=4) = 1365 \cdot (0.18)^4 \cdot (0.82)^{11} \approx 1365 \cdot 0.00104976 \cdot 0.1165609 \approx \textbf{0.167}$ So, there's about a 16.7% chance of exactly 4 out of 15 people working out daily.

STEP 7

We want to find $P(z < 0.47)$ .
Using a z-table or calculator, we find this probability to be approximately $\textbf{0.6808}$ .

STEP 8

We want $P(z > -1.83)$ .
This is the same as $1 - P(z < -1.83)$ .
Using a z-table or calculator, $P(z < -1.83) \approx 0.0336$ , so $P(z > -1.83) \approx 1 - 0.0336 = \textbf{0.9664}$ .

STEP 9

We're looking for $P(-1.29 < z < 0.64)$ .
This is equal to $P(z < 0.64) - P(z < -1.29)$ .
Using a z-table or calculator, $P(z < 0.64) \approx 0.7389$ and $P(z < -1.29) \approx 0.0985$ .
So, $P(-1.29 < z < 0.64) \approx 0.7389 - 0.0985 = \textbf{0.6404}$ .

STEP 10

We want $P(-1.96 < z < 1.96)$ .
This is $P(z < 1.96) - P(z < -1.96)$ .
From a z-table or calculator, $P(z < 1.96) \approx 0.9750$ and $P(z < -1.96) \approx 0.0250$ .
Therefore, $P(-1.96 < z < 1.96) \approx 0.9750 - 0.0250 = \textbf{0.95}$ .

STEP 11

The probability of exactly 4 out of 15 people working out daily is approximately $\textbf{0.167}$ .
For the standard normal distribution: a) $P(z < 0.47) \approx \textbf{0.6808}$ , b) $P(z > -1.83) \approx \textbf{0.9664}$ , c) $P(-1.29 < z < 0.64) \approx \textbf{0.6404}$ , and d) $P(-1.96 < z < 1.96) \approx \textbf{0.95}$ .

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Binomial Probability
2. Z-score Less Than
3. Z-score Greater Than
4. Z-score Between Two Values (1)
5. Z-score Between Two Values (2)

STEP 3

The binomial probability formula tells us the chance of getting exactly $k$ successes in $n$ trials: $P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{(n-k)}$ Where $p$ is the probability of success on a single trial.

STEP 4

We have $n = \textbf{15}$ people (trials), we want $k = \textbf{4}$ people to work out (successes), and the probability of someone working out daily is $p = \textbf{0.18}$ .
Let's plug these into our formula! $P(X=4) = \binom{15}{4} \cdot (0.18)^4 \cdot (1-0.18)^{(15-4)}$

STEP 5

The binomial coefficient $\binom{15}{4}$ represents the number of ways to choose 4 people out of 15.
It's calculated as: $\binom{15}{4} = \frac{15!}{4! \cdot (15-4)!} = \frac{15!}{4! \cdot 11!} = \frac{15 \cdot 14 \cdot 13 \cdot 12}{4 \cdot 3 \cdot 2 \cdot 1} = \textbf{1365}$

STEP 6

Now, let's plug everything back into our binomial formula: $P(X=4) = 1365 \cdot (0.18)^4 \cdot (0.82)^{11} \approx 1365 \cdot 0.00104976 \cdot 0.1165609 \approx \textbf{0.167}$ So, there's about a 16.7% chance of exactly 4 out of 15 people working out daily.

STEP 7

We want to find $P(z < 0.47)$ .
Using a z-table or calculator, we find this probability to be approximately $\textbf{0.6808}$ .

STEP 8

We want $P(z > -1.83)$ .
This is the same as $1 - P(z < -1.83)$ .
Using a z-table or calculator, $P(z < -1.83) \approx 0.0336$ , so $P(z > -1.83) \approx 1 - 0.0336 = \textbf{0.9664}$ .

STEP 9

We're looking for $P(-1.29 < z < 0.64)$ .
This is equal to $P(z < 0.64) - P(z < -1.29)$ .
Using a z-table or calculator, $P(z < 0.64) \approx 0.7389$ and $P(z < -1.29) \approx 0.0985$ .
So, $P(-1.29 < z < 0.64) \approx 0.7389 - 0.0985 = \textbf{0.6404}$ .

STEP 10

We want $P(-1.96 < z < 1.96)$ .
This is $P(z < 1.96) - P(z < -1.96)$ .
From a z-table or calculator, $P(z < 1.96) \approx 0.9750$ and $P(z < -1.96) \approx 0.0250$ .
Therefore, $P(-1.96 < z < 1.96) \approx 0.9750 - 0.0250 = \textbf{0.95}$ .

STEP 11

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. Binomial Probability2. Z-score Less Than3. Z-score Greater Than4. Z-score Between Two Values (1)5. Z-score Between Two Values (2)

STEP 3

The binomial probability formula tells us the chance of getting *exactly* kkk successes in nnn trials: P(X=k)=(nk)⋅pk⋅(1−p)(n−k)P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{(n-k)}P(X=k)=(kn​)⋅pk⋅(1−p)(n−k) Where ppp is the probability of success on a single trial.

STEP 4

STEP 5

STEP 6

STEP 7

We want to find P(z<0.47)P(z < 0.47)P(z<0.47).Using a z-table or calculator, we find this probability to be approximately 0.6808\textbf{0.6808}0.6808.

STEP 8

STEP 9

STEP 10

STEP 11

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. Binomial Probability
2. Z-score Less Than
3. Z-score Greater Than
4. Z-score Between Two Values (1)
5. Z-score Between Two Values (2)

The binomial probability formula tells us the chance of getting exactly $k$ successes in $n$ trials: $P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{(n-k)}$ Where $p$ is the probability of success on a single trial.

We want to find $P(z < 0.47)$ .
Using a z-table or calculator, we find this probability to be approximately $\textbf{0.6808}$ .