Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

$X$ is a normally distributed random variable with mean $10$ and standard deviation $3$ . What is the probability that $X$ is between $9$ and $11$ ? Write your answer as a decimal rounded to the nearest thousandth.

View step-by-step help

View step-by-step help

Find probabilities using the normal distribution I

Full solution

Q.

X

is a normally distributed random variable with mean

10

and standard deviation

3

. What is the probability that

X

is between

9

and

11

? Write your answer as a decimal rounded to the nearest thousandth.

Calculate z-score for $X=9$ : Mean ( $\mu$ ) is $10$ , standard deviation ( $\sigma$ ) is $3$ . Calculate the z-score for $X=9$ . $\newline$ $Z = \frac{X - \mu}{\sigma} = \frac{9 - 10}{3} = -\frac{1}{3}$ .
Calculate z-score for $X=11$ : Calculate the z-score for $X=11$ . $Z = \frac{X - \mu}{\sigma} = \frac{11 - 10}{3} = \frac{1}{3}$ .
Probability within $1$ standard deviation: Using the $0.68$ - $0.95$ - $0.997$ rule, the probability that $X$ is within $1$ standard deviation ( $\mu \pm \sigma$ ) is about $0.68$ .
Probability between $9$ and $11$ : Since $9$ and $11$ are within $1$ standard deviation from the mean, the probability that $X$ is between $9$ and $11$ is less than $0.68$ .
Estimate exact probability: The exact probability for z-scores $-\frac{1}{3}$ and $\frac{1}{3}$ is not given by the $0.68$ - $0.95$ - $0.997$ rule, but we can estimate it to be roughly half of $0.68$ , because the interval from $9$ to $11$ is about half the interval from $\mu - \sigma$ to $\mu + \sigma$ .

More problems from Find probabilities using the normal distribution I

Question

A restaurant server claims that \(28\)% of the time he leaves candy with the bill, the customer tips well.

\newline

If the server's claim is true, and one day he leaves candy with \(4\) customers' bills, what is the probability that exactly \(3\) of those customers will tip well?

\newline

Write your answer as a decimal rounded to the nearest thousandth.

Posted 1 month ago

Question

There is a spinner with `50` equally likely sections, numbered from `1` to `50`. You have the opportunity to spin it. If the number is odd, you win $`11`. If the number is even, you win nothing. If you play the game, what is the expected payoff?\(\$\)_____

Posted 2 months ago

Question

There are two different raffles you can enter.

\newline

In raffle A, one ticket will win a `$720` prize, and the other tickets will win nothing. There are

1,000

in the raffle, each costing `$11`.

\newline

Raffle B is for a `$370` prize. Out of

125

tickets, each costing `$5`, one ticket will win the prize, and the other tickets will win nothing.

\newline

Which raffle is a better deal?

\newline

\newline

\text{[A]Raffle A}

\newline

\text{[B]Raffle B}

Posted 2 months ago

Question

X

is a normally distributed random variable with mean

49

and standard deviation

25

\newline

What is the probability that

X

24

74

?

\newline

0.68-0.95-0.997

rule and write your answer as a decimal. Round to the nearest thousandth if necessary.

\newline

Posted 2 months ago

Question

X

is a normally distributed random variable with mean

65

and standard deviation

8

\newline

What is the probability that

X

62

68

?

\newline

Write your answer as a decimal rounded to the nearest thousandth.

\newline

Posted 2 months ago

Question

Complete the statement. Round your answer to the nearest thousandth.

\newline

In a population that is normally distributed with mean

43

and standard deviation

3

30\%

of the values are those less than ______.

Posted 2 months ago

Question

Y

is the number of desserts available in a randomly chosen restaurant.

\newline

Is the random variable

Y

discrete or continuous?

\newline

\newline

\text{[A]discrete}

\newline

\text{[B]continuous}

Posted 2 months ago

Question

Which of the following functions are continuous for all real numbers?

\newline

g(x)=\ln (x)

\newline

f(x)=\frac{1}{x}

\newline

1

\newline

g

\newline

f

\newline

g

f

\newline

g

f

Posted 3 months ago

Question

Which of the following functions are continuous at

x=-1

\newline

g(x)=\sin (x+1)

\newline

f(x)=\frac{1}{x-1}

\newline

1

\newline

g

\newline

f

\newline

g

f

\newline

g

f

Posted 2 months ago

Question

g

be a continuous function on the closed interval

[-1,4]

g(-1)=-4

g(4)=1

\newline

Which of the following is guaranteed by the Intermediate Value Theorem?

\newline

1

\newline

g(c)=-3

for at least one

c

-1

4

\newline

g(c)=3

for at least one

c

-4

1

\newline

g(c)=-3

for at least one

c

-4

1

\newline

g(c)=3

for at least one

c

-1

4

Posted 1 month ago