Assume that there is a sampling distribution of \bar{x} of size 34 and they are selected at random from a normally distributed with a mean of 45 and a standard deviation of 6.2. Find the probability that 1 randomly selected person has a value less than 40.

Question
Random variables
asked 2021-02-25
Assume that there is a sampling distribution of \(\displaystyle\overline{{{x}}}\) of size 34 and they are selected at random from a normally distributed with a mean of 45 and a standard deviation of 6.2.
Find the probability that 1 randomly selected person has a value less than 40.

Answers (1)

2021-02-26
Here, \(\displaystyle{X}\sim{N}{\left({45},{6.2}\right)}\).
The probability that a randomly selected person has a value less than 40 is calculated as follows:
\(\displaystyle{P}{\left({X}{<}{40}\right)}={P}{\left({\frac{{{X}-\mu}}{{\sigma}}}{<}{\frac{{{40}-{45}}}{{{6.2}}}}\right)}\)</span>
\(\displaystyle={P}{\left({z}{<}-{0.81}\right)}{\left[{F}{r}{o}{m}\ {t}{h}{e}\ {s}{\tan{{d}}}{a}{r}{d}\ {\left\|{a}\right\|}{l}\ {t}{a}{b}\le,\ {P}{\left({z}{<}-{0.81}\right)}={0.2090}\right]}={0.2090}\)</span>
The probability that a randomly selected person has a value less than 40 is 0.2090.
0

Relevant Questions

asked 2020-10-23
1. Find each of the requested values for a population with a mean of \(? = 40\), and a standard deviation of \(? = 8\) A. What is the z-score corresponding to \(X = 52?\) B. What is the X value corresponding to \(z = - 0.50?\) C. If all of the scores in the population are transformed into z-scores, what will be the values for the mean and standard deviation for the complete set of z-scores? D. What is the z-score corresponding to a sample mean of \(M=42\) for a sample of \(n = 4\) scores? E. What is the z-scores corresponding to a sample mean of \(M= 42\) for a sample of \(n = 6\) scores? 2. True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1 3. Interpret the location, direction, and distance (near or far) of the following zscores: \(a. -2.00 b. 1.25 c. 3.50 d. -0.34\) 4. You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with \(\mu = 78\) and \(\sigma = 12\). Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: \(82, 74, 62, 68, 79, 94, 90, 81, 80\). 5. You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about \($12 (\mu = 42, \sigma = 12)\). You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is $44.50 from tips. Test for a difference between this value and the population mean at the \(\alpha = 0.05\) level of significance.
asked 2020-11-29
CNNBC recently reported that the mean annual cost of auto insurance is 954 dollars. Assume the standard deviation is 271 dollars. You take a simple random sample of 96 auto insurance policies.
Find the probability that a sample of size \(\displaystyle{n}={96}\) is randomly selected with a mean less than 969 dollars.
\(\displaystyle{P}{\left(\overline{{{x}}}{<}{969}\right)}={P}{\left(\overline{{{x}}}{<}{969}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.
asked 2021-02-13
A population of values has a normal distribution with \(\displaystyle\mu={192.6}\) and \(\displaystyle\sigma={34.4}\). You intend to draw a random sample of size \(\displaystyle{n}={173}\).
Find the probability that a single randomly selected value is less than 186.1.
\(\displaystyle{P}{\left({X}{<}{186.1}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.
asked 2021-02-25
A population of values has a normal distribution with \(\displaystyle\mu={192.6}\) and \(\displaystyle\sigma={34.4}\). You intend to draw a random sample of size \(\displaystyle{n}={173}\).
Find the probability that a sample of size \(\displaystyle{n}={173}\) is randomly selected with a mean less than 186.1.
\(\displaystyle{P}{\left({M}{<}{186.1}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.
asked 2021-01-10
A population of values has a normal distribution with mean =136.4 and standard deviation =30.2. A random sample of size \(\displaystyle{n}={158}\) is drawn.
Find the probability that a single randomly selected value is greater than 135. Roung your answer to four decimal places.
\(\displaystyle{P}{\left({X}{>}{135}\right)}=\)?
asked 2021-01-31
CNBC recently reported that the mean annual cost of auto insurance is $998. Assume the standard deviation is $298.
Find the probability that a sample of size 61 is randomly selected with a mean less than $985.
\(\displaystyle{P}{\left(\overline{{{X}}}{<}{985}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.
asked 2021-01-22
A population of values has a normal distribution with mean = 37.4 and standard deviation 77.4. If a random sample of size \(\displaystyle{n}={15}\) is selected,
Find the probability that a single randomly selected value is greater than 53.4. Round your answer to four decimals.
\(\displaystyle{P}{\left({X}{>}{53.4}\right)}=\)?
asked 2021-02-08
A population of values has a normal distribution with \(\displaystyle\mu={204.3}\) and \(\displaystyle\sigma={43}\). You intend to draw a random sample of size \(\displaystyle{n}={111}\).
Find the probability that a single randomly selected value is less than 191.2.
\(\displaystyle{P}{\left({X}{<}{191.2}\right)}=\)?
Find the probability that a sample of size \(\displaystyle{n}={111}\) is randomly selected with a mean less than 191.2.
\(\displaystyle{P}{\left({M}{<}{191.2}\right)}=\)?
Write your answers as numbers accurate to 4 decimal places.
asked 2020-11-08
A population of values has a normal distribution with mean 18.6 and standard deviation 57. If a random sample of size \(\displaystyle{n}={25}\) is selected,
Find the probability that a sample of size \(\displaystyle{n}={25}\) is randomly selected with a mean greater than 17.5 round your answer to four decimal places.
P=?
asked 2021-01-27
A population of values has a normal distribution with mean 191.4 and standard deviation of 69.7. A random sample of size \(\displaystyle{n}={153}\) is drawn.
Find the probability that a single randomly selected value is between 188 and 206.6 round your answer to four decimal places.
P=?
...