Math / Data & Statistics

QuestionSAT scores: Assume that in a given year the mean mathematics SAT score was 605 , and the standard deviation was 136. A sample of 76 scores is chosen. Use Excel.
Part: $0 / 5$
Part 1 of 5 (a) What is the probability that the sample mean score is less than 589 ? Round the answer to at least four decimal places.
The probability that the sample mean score is less than 589 is $\square$

Studdy Solution

STEP 1

What is this asking? If we grab 76 random SAT math scores, what are the chances the average of those scores is less than 589? Watch out! We're dealing with the sample mean, not individual scores, so the standard deviation changes!

STEP 2

1. Calculate the standard error.
2. Calculate the z-score.
3. Find the probability.

STEP 3

Alright, so we're dealing with a sample of scores, not the entire population.
This means we need the standard error, which tells us how spread out the sample means are likely to be.
It's like the standard deviation, but for sample means!

STEP 4

The formula for the standard error is $SE = \frac{\sigma}{\sqrt{n}}$ , where $\sigma$ is the population standard deviation ( $\sigma = 136$ ) and $n$ is the sample size ( $n = 76$ ).

STEP 5

Let's plug in the numbers: $SE = \frac{136}{\sqrt{76}} \approx \frac{136}{8.718} \approx 15.60$ .
So, our standard error is approximately $15.60$ .

STEP 6

The z-score tells us how far away our sample mean is from the population mean, in terms of standard errors.
It's like a standardized score that lets us use the z-table to find probabilities.

STEP 7

The formula for the z-score is $z = \frac{\bar{x} - \mu}{SE}$ , where $\bar{x}$ is the sample mean we're interested in ( $\bar{x} = 589$ ), $\mu$ is the population mean ( $\mu = 605$ ), and $SE$ is the standard error we just calculated ( $SE \approx 15.60$ ).

STEP 8

Plugging in the values, we get $z = \frac{589 - 605}{15.60} = \frac{-16}{15.60} \approx -1.03$ .
Our z-score is approximately $-1.03$ .

STEP 9

Now, we can use our z-score to find the probability that the sample mean is less than 589.
We'll use the z-table (or a calculator) to find the area to the left of our z-score of $-1.03$ .

STEP 10

Looking up $-1.03$ in the z-table, we find a probability of approximately $0.1515$ .

STEP 11

The probability that the sample mean score is less than 589 is approximately $0.1515$ .

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Studdy Solution

STEP 1

What is this asking? If we grab 76 random SAT math scores, what are the chances the average of those scores is less than 589? Watch out! We're dealing with the sample mean, not individual scores, so the standard deviation changes!

STEP 2

1. Calculate the standard error.
2. Calculate the z-score.
3. Find the probability.

STEP 3

Alright, so we're dealing with a sample of scores, not the entire population.
This means we need the standard error, which tells us how spread out the sample means are likely to be.
It's like the standard deviation, but for sample means!

STEP 4

The formula for the standard error is $SE = \frac{\sigma}{\sqrt{n}}$ , where $\sigma$ is the population standard deviation ( $\sigma = 136$ ) and $n$ is the sample size ( $n = 76$ ).

STEP 5

Let's plug in the numbers: $SE = \frac{136}{\sqrt{76}} \approx \frac{136}{8.718} \approx 15.60$ .
So, our standard error is approximately $15.60$ .

STEP 6

The z-score tells us how far away our sample mean is from the population mean, in terms of standard errors.
It's like a standardized score that lets us use the z-table to find probabilities.

STEP 7

The formula for the z-score is $z = \frac{\bar{x} - \mu}{SE}$ , where $\bar{x}$ is the sample mean we're interested in ( $\bar{x} = 589$ ), $\mu$ is the population mean ( $\mu = 605$ ), and $SE$ is the standard error we just calculated ( $SE \approx 15.60$ ).

STEP 8

Plugging in the values, we get $z = \frac{589 - 605}{15.60} = \frac{-16}{15.60} \approx -1.03$ .
Our z-score is approximately $-1.03$ .

STEP 9

Now, we can use our z-score to find the probability that the sample mean is less than 589.
We'll use the z-table (or a calculator) to find the area to the left of our z-score of $-1.03$ .

STEP 10

Looking up $-1.03$ in the z-table, we find a probability of approximately $0.1515$ .

STEP 11

The probability that the sample mean score is less than 589 is approximately $0.1515$ .

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Studdy Solution

STEP 1

What is this asking? If we grab 76 random SAT math scores, what are the chances the average of *those* scores is less than 589? Watch out! We're dealing with the *sample mean*, not individual scores, so the standard deviation changes!

STEP 2

1. Calculate the standard error.2. Calculate the z-score.3. Find the probability.

STEP 3

Alright, so we're dealing with a *sample* of scores, not the entire population.This means we need the **standard error**, which tells us how spread out the *sample means* are likely to be.It's like the standard deviation, but for sample means!

STEP 4

The formula for the standard error is SE=σnSE = \frac{\sigma}{\sqrt{n}}SE=n​σ​, where σ\sigmaσ is the **population standard deviation** (σ=136\sigma = 136σ=136) and nnn is the **sample size** (n=76n = 76n=76).

STEP 5

Let's plug in the numbers: SE=13676≈1368.718≈15.60SE = \frac{136}{\sqrt{76}} \approx \frac{136}{8.718} \approx 15.60SE=76​136​≈8.718136​≈15.60.So, our **standard error** is approximately 15.6015.6015.60.

STEP 6

The **z-score** tells us how far away our sample mean is from the population mean, in terms of standard errors.It's like a standardized score that lets us use the z-table to find probabilities.

STEP 7

STEP 8

Plugging in the values, we get z=589−60515.60=−1615.60≈−1.03z = \frac{589 - 605}{15.60} = \frac{-16}{15.60} \approx -1.03z=15.60589−605​=15.60−16​≈−1.03.Our **z-score** is approximately −1.03-1.03−1.03.

STEP 9

Now, we can use our **z-score** to find the probability that the sample mean is less than 589.We'll use the z-table (or a calculator) to find the area to the *left* of our z-score of −1.03-1.03−1.03.

STEP 10

Looking up −1.03-1.03−1.03 in the z-table, we find a probability of approximately 0.15150.15150.1515.

STEP 11

The probability that the sample mean score is less than 589 is approximately 0.15150.15150.1515.

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.

What is this asking? If we grab 76 random SAT math scores, what are the chances the average of those scores is less than 589? Watch out! We're dealing with the sample mean, not individual scores, so the standard deviation changes!

1. Calculate the standard error.
2. Calculate the z-score.
3. Find the probability.

Alright, so we're dealing with a sample of scores, not the entire population.
This means we need the standard error, which tells us how spread out the sample means are likely to be.
It's like the standard deviation, but for sample means!

The formula for the standard error is $SE = \frac{\sigma}{\sqrt{n}}$ , where $\sigma$ is the population standard deviation ( $\sigma = 136$ ) and $n$ is the sample size ( $n = 76$ ).

Let's plug in the numbers: $SE = \frac{136}{\sqrt{76}} \approx \frac{136}{8.718} \approx 15.60$ .
So, our standard error is approximately $15.60$ .

The z-score tells us how far away our sample mean is from the population mean, in terms of standard errors.
It's like a standardized score that lets us use the z-table to find probabilities.

Plugging in the values, we get $z = \frac{589 - 605}{15.60} = \frac{-16}{15.60} \approx -1.03$ .
Our z-score is approximately $-1.03$ .

Now, we can use our z-score to find the probability that the sample mean is less than 589.
We'll use the z-table (or a calculator) to find the area to the left of our z-score of $-1.03$ .

Looking up $-1.03$ in the z-table, we find a probability of approximately $0.1515$ .

The probability that the sample mean score is less than 589 is approximately $0.1515$ .