Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

A study by the department of education of a certain state was trying to determine the mean SAT scores of the graduating high school seniors. The study examined the scores of a random sample of 98 graduating seniors and found the mean score to be 492 with a standard deviation of 117 . At the
95% confidence level, use the normal distribution/empirical rule to estimate the margin of error for the mean, rounding to the nearest tenth. (Do not write
+- ).
Answer:

A study by the department of education of a certain state was trying to determine the mean SAT scores of the graduating high school seniors. The study examined the scores of a random sample of $98$ graduating seniors and found the mean score to be $492$ with a standard deviation of $117$ . At the $95 \%$ confidence level, use the normal distribution/empirical rule to estimate the margin of error for the mean, rounding to the nearest tenth. (Do not write $\pm$ ). $\newline$ Answer:

View step-by-step help

View step-by-step help

Use normal distributions to approximate binomial distributions

Full solution

Q. A study by the department of education of a certain state was trying to determine the mean SAT scores of the graduating high school seniors. The study examined the scores of a random sample of

98

graduating seniors and found the mean score to be

492

with a standard deviation of

117

. At the

95 \%

confidence level, use the normal distribution/empirical rule to estimate the margin of error for the mean, rounding to the nearest tenth. (Do not write

\pm

).

\newline

Answer:

Identify Z-score: To calculate the margin of error at the $95$ % confidence level using the normal distribution, we need to use the formula for the margin of error (ME) which is: $\newline$ ME = $Z \times (\sigma/\sqrt{n})$ $\newline$ where $Z$ is the Z-score corresponding to the desired confidence level, $\sigma$ is the standard deviation, and $n$ is the sample size.
Calculate Margin of Error Formula: First, we need to find the Z-score that corresponds to the $95$ % confidence level. For a $95$ % confidence level in a normal distribution, the Z-score is approximately $1.96$ . This value can be found in Z-score tables or using a standard normal distribution calculator.
Plug in Values: Next, we plug in the values we have into the margin of error formula: $\newline$ $\sigma = 117$ (standard deviation) $\newline$ $n = 98$ (sample size) $\newline$ $Z = 1.96$ (Z-score for $95$ % confidence) $\newline$ $ME = 1.96 \times (117/\sqrt{98})$
Calculate Margin of Error: Now we calculate the margin of error: $\newline$ ME = $1.96 \times (\frac{117}{\sqrt{98}})$ $\newline$ ME = $1.96 \times (\frac{117}{9.8995})$ (rounded $\sqrt{98}$ to four decimal places) $\newline$ ME = $1.96 \times 11.8182$ (rounded $\frac{117}{9.8995}$ to four decimal places) $\newline$ ME = $23.1645$ (rounded to four decimal places)
Round Margin of Error: Finally, we round the margin of error to the nearest tenth: $ME \approx 23.2$

More problems from Use normal distributions to approximate binomial distributions

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 3 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 3 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 3 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 3 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 3 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 2 months ago