# 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 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. 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.

### Relevant Questions 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. 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.
$$\displaystyle{P}{\left(\overline{{{x}}}{<}{969}\right)}={P}{\left(\overline{{{x}}}{<}{969}\right)}=$$? 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.
$$\displaystyle{P}{\left({X}{<}{186.1}\right)}=$$? 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.
$$\displaystyle{P}{\left({M}{<}{186.1}\right)}=$$? 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.
$$\displaystyle{P}{\left({X}{>}{135}\right)}=$$? 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.
$$\displaystyle{P}{\left(\overline{{{X}}}{<}{985}\right)}=$$? 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.
$$\displaystyle{P}{\left({X}{>}{53.4}\right)}=$$? 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.
$$\displaystyle{P}{\left({M}{<}{191.2}\right)}=$$? 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. 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.