Statistics Informed Decisions Using Data

Michael Sullivan III

Chapter 8

Sampling Distributions - all with Video Answers

Educators

Section 1

Distribution of the Sample Mean

Problem 1

The ________ _________ of the sample mean, $\bar{x}$, is the probability distribution of all possible values of the random variable $\bar{x}$ computed from a sample of size $n$ from a population with mean $\mu$ and standard deviation $\sigma .$

Diane Koenig

Diane Koenig

Numerade Educator

Problem 2

Suppose a simple random sample of size $n$ is drawn from a large population with mean $\mu$ and standard deviation $\sigma .$ The sampling distribution of $\bar{x}$ has mean $\mu_{\bar{x}}=$ ______ and standard deviation $\sigma_{\bar{x}}=$ ______.

Diane Koenig

Numerade Educator

Problem 4

True or False: The distribution of the sample mean, $\bar{x},$ will be normally distributed if the sample is obtained from a population that is normally distributed, regardless of the sample size.

Carly Stoner

Numerade Educator

Problem 5

True or False: The distribution of the sample mean, $\bar{x},$ will be normally distributed if the sample is obtained from a population that is not normally distributed, regardless of the sample size.

Carly Stoner

Numerade Educator

Problem 6

True or False: To cut the standard error of the mean in half, the sample size must be doubled.

Emily Himsel

Numerade Educator

Problem 7

A simple random sample of size $n=10$ is obtained from a population that is normally distributed with $\mu=30$ and $\sigma=8$. What is the sampling distribution of $\bar{x} ?$

Diane Koenig

Numerade Educator

Problem 8

A simple random sample of size $n=40$ is obtained from a population with $\mu=50$ and $\sigma=4 .$ Does the population need to be normally distributed for the sampling distribution of $\bar{x}$ to be approximately normally distributed? Why? What is the sampling distribution of $\bar{x} ?$

Diane Koenig

Numerade Educator

Problem 9

Determine $\mu_{\bar{x}}$ and $\sigma_{\bar{x}}$ from the given parameters of the population and the sample size.
$\mu=80, \sigma=14, n=49$

Jon Southam

Numerade Educator

Problem 10

Determine $\mu_{\bar{x}}$ and $\sigma_{\bar{x}}$ from the given parameters of the population and the sample size.
$\mu=64, \sigma=18, n=36$

Diane Koenig

Numerade Educator

Problem 11

Determine $\mu_{\bar{x}}$ and $\sigma_{\bar{x}}$ from the given parameters of the population and the sample size.
$\mu=52, \sigma=10, n=21$

Diane Koenig

Numerade Educator

Problem 12

Determine $\mu_{\bar{x}}$ and $\sigma_{\bar{x}}$ from the given parameters of the population and the sample size.
$\mu=27, \sigma=6, n=15$

Diane Koenig

Numerade Educator

Problem 13

Answer the following questions for the sampling distribution of the sample mean shown to the right.
(a) What is the value of $\mu_{\bar{x}}$ ?
(b) What is the value of $\sigma_{\bar{x}}$ ?
(c) If the sample size is $n=16$, what is likely true about the shape of the population?
(d) If the sample size is $n=16,$ what is the standard deviation of the population from which the sample was drawn?

Sheryl Ezze

Numerade Educator

Problem 14

Answer the following questions for the sampling distribution of the sample mean shown to the right.
(a) What is the value of $\mu_{\bar{x}}$ ?
(b) What is the value of $\sigma_{\bar{x}}$ ?
(c) If the sample size is $n=9$ what is likely true about the shape of the population?
(d) If the sample size is $n=9$, what is the standard deviation of the population from which the sample was drawn?

Sheryl Ezze

Numerade Educator

Problem 15

A simple random sample of size $n=49$ is obtained from a population with $\mu=80$ and $\sigma=14$
(a) Describe the sampling distribution of $\bar{x}$.
(b) What is $P(\bar{x}>83) ?$
(c) What is $P(\bar{x} \leq 75.8) ?$
(d) What is $P(78.3<\bar{x}<85.1) ?$

Diane Koenig

Numerade Educator

Problem 16

A simple random sample of size $n=36$ is obtained from a population with $\mu=64$ and $\sigma=18$.
(a) Describe the sampling distribution of $\bar{x}$.
(b) What is $P(\bar{x}<62.6) ?$
(c) What is $P(\bar{x} \geq 68.7) ?$
(d) What is $P(59.8<\bar{x}<65.9) ?$

Sneha Ravi

Numerade Educator

Problem 17

A simple random sample of size $n=12$ is obtained from a population with $\mu=64$ and $\sigma=17$
(a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample mean? Assuming that this condition is true, describe the sampling distribution of $\bar{x}$.
(b) Assuming that the requirements described in part (a) are satisfied, determine $P(\bar{x}<67.3)$
(c) Assuming that the requirements described in part (a) are satisfied, determine $P(\bar{x} \geq 65.2)$.

Neel Faucher

Numerade Educator

Problem 18

A simple random sample of size $n=20$ is obtained from a population with $\mu=64$ and $\sigma=17$
(a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample mean? Assuming that this condition is true, describe the sampling distribution of $\bar{x}$.
(b) Assuming that the requirements described in part (a) are satisfied, determine $P(\bar{x}<67.3)$
(c) Assuming that the requirements described in part (a) are satisfied, determine $P(\bar{x} \geq 65.2)$
(d) Compare the results obtained in parts (b) and (c) with the results obtained in parts (b) and (c) in Problem 17 . What effect does increasing the sample size have on the probabilities? Why do you think this is the case?

Sneha Ravi

Numerade Educator

Problem 19

The length of human pregnancies is approximately normally distributed with mean $\mu=266$ days and standard deviation $\sigma=16$ days.
(a) What is the probability a randomly selected pregnancy lasts less than 260 days?
(b) Suppose a random sample of 20 pregnancies is obtained. Describe the sampling distribution of the sample mean length of human pregnancies.
(c) What is the probability that a random sample of 20 pregnancies has a mean gestation period of 260 days or less?
(d) What is the probability that a random sample of 50 pregnancies has a mean gestation period of 260 days or less?
(e) What might you conclude if a random sample of 50 pregnancies resulted in a mean gestation period of 260 days or less?
(f) What is the probability a random sample of size 15 will have a mean gestation period within 10 days of the mean?

Sneha Ravi

Numerade Educator

Problem 20

The upper leg length of 20 - to 29 -year-old males is normally distributed with a mean length of $43.7 \mathrm{~cm}$ and a standard deviation of $4.2 \mathrm{~cm} .$ Source: "Anthropometric Reference Data for Children and Adults: U.S. Population, 1999-2002"; Volume 361, July 7, 2005.
(a) What is the probability that a randomly selected 20 - to 29 -yearold male has an upper leg length that is less than $40 \mathrm{~cm} ?$
(b) A random sample of 9 males who are 20 to 29 years old is obtained. What is the probability that the mean upper leg length is less than $40 \mathrm{~cm} ?$
(c) What is the probability that a random sample of 12 males who are $20-29$ years old results in a mean upper leg length that is less than $40 \mathrm{~cm} ?$
(d) What effect does increasing the sample size have on the probability? Provide an explanation for this result.
(e) A random sample of 15 males who are $20-29$ years old results in a mean upper leg length of $46 \mathrm{~cm} .$ Do you find this result unusual? Why?

Sheryl Ezze

Numerade Educator

Problem 21

The reading speed of second grade students is approximately normal, with a mean of 90 words per minute (wpm) and a standard deviation of 10 wpm.
(a) What is the probability a randomly selected student will read more than 95 words per minute?
(b) What is the probability that a random sample of 12 second grade students results in a mean reading rate of more than 95 words per minute?
(c) What is the probability that a random sample of 24 second grade students results in a mean reading rate of more than 95 words per minute?
(d) What effect does increasing the sample size have on the probability? Provide an explanation for this result.
(e) A teacher instituted a new reading program at school. After 10 weeks in the program, it was found that the mean reading speed of a random sample of 20 second grade students was 92.8 wpm. What might you conclude based on this result?
(f) There is a $5 \%$ chance that the mean reading speed of a random sample of 20 second grade students will exceed what value?

Carson Merrill

Numerade Educator

Problem 22

The most famous geyser in the world, Old Faithful in Yellowstone National Park, has a mean time between eruptions of 85 minutes. If the interval of time between eruptions is normally distributed with standard deviation 21.25 minutes, answer the following questions:
Source: www.unmuseum.org
(a) What is the probability that a randomly selected time interval between eruptions is longer than 95 minutes?
(b) What is the probability that a random sample of 20 time intervals between eruptions has a mean longer than 95 minutes?
(c) What is the probability that a random sample of 30 time intervals between eruptions has a mean longer than 95 minutes?
(d) What effect does increasing the sample size have on the probability? Provide an explanation for this result.
(e) What might you conclude if a random sample of 30 time intervals between eruptions has a mean longer than 95 minutes?
(f) On a certain day, suppose there are 22 time intervals for Old Faithful. Treating these 22 eruptions as a random sample, the likelihood the mean length of time between eruptions exceeds __________ minutes is 0.20.

Sheryl Ezze

Numerade Educator

Problem 23

The S\&P 500 is a collection of 500 stocks of publicly traded companies. Using data obtained from Yahoo! Finance, the monthly rates of return of the S\&P500 since 1950 are normally distributed. The mean rate of return is $0.007233(0.7233 \%),$ and the standard deviation for rate of return is $0.04135(4.135 \%)$.
(a) What is the probability that a randomly selected month has a positive rate of return? That is, what is $P(x>0) ?$
(b) Treating the next 12 months as a simple random sample, what is the probability that the mean monthly rate of return will be positive? That is, with $n=12,$ what is $P(\bar{x}>0) ?$
(c) Treating the next 24 months as a simple random sample, what is the probability that the mean monthly rate of return will be positive?
(d) Treating the next 36 months as a simple random sample, what is the probability that the mean monthly rate of return will be positive?
(e) Use the results of parts (b)-(d) to describe the likelihood of earning a positive rate of return on stocks as the investment time horizon increases.

Sheryl Ezze

Numerade Educator

Problem 24

A very good poker player is expected to earn $\$ 1$ per hand in $\$ 100 / \$ 200$ Texas Hold'em. The standard deviation is approximately $\$ 32 .$
(a) What is the probability a very good poker player earns a profit (more than \$0) after playing 50 hands in $\$ 100 / \$ 200$ Texas Hold'em?
(b) What is the probability a very good poker player loses (earns less than \$0) after playing 100 hands in $\$ 100 / \$ 200$ Texas Hold'em?
(c) What proportion of the time can a very good poker player expect to earn at least $\$ 500$ after playing 100 hands in $\$ 100 / \$ 200$ Texas Hold'em? Hint: $\$ 500$ after 100 hands is a mean of $\$ 5$ per hand.
(d) Would it be unusual for a very good poker player to lose at least $\$ 1000$ after playing 100 hands in $\$ 100 / \$ 200$ Texas Hold'em?
(e) Suppose twenty hands are played per hour. What is the probability that a very good poker player earns a profit during a twenty-four hour marathon session?

Sheryl Ezze

Numerade Educator

Problem 25

The shape of the distribution of the time required to get an oil change at a 10 -minute oil-change facility is unknown. However, records indicate that the mean time for an oil change is 11.4 minutes, and the standard deviation for oilchange time is 3.2 minutes.
(a) To compute probabilities regarding the sample mean using the normal model, what size sample would be required?
(b) What is the probability that a random sample of $n=40$ oil changes results in a sample mean time of less than 10 minutes?
(c) Suppose the manager agrees to pay each employee a $\$ 50$ bonus if they meet a certain goal. On a typical Saturday, the oil-change facility will perform 40 oil changes between 10 A.M. and 12 P.M. Treating this as a random sample, what mean oil-change time would there be a $10 \%$ chance of being at or below? This will be the goal established by the manager.

Sheryl Ezze

Numerade Educator

Problem 26

The quality-control manager of a Long John Silver's restaurant wants to analyze the length of time that a car spends at the drive-through window waiting for an order. It is determined that the mean time spent at the window is 59.3 seconds with a standard deviation of
13.1 seconds. The distribution of time at the window is skewed right (data based on information provided by Danica Williams, student at Joliet Junior College).
(a) To obtain probabilities regarding a sample mean using the normal model, what size sample is required?
(b) The quality-control manager wishes to use a new delivery system designed to get cars through the drive-through system faster. A random sample of 40 cars results in a sample mean time spent at the window of 56.8 seconds. What is the probability of obtaining a sample mean of 56.8 seconds or less, assuming that the population mean is 59.3 seconds? Do you think that the new system is effective?
(c) Treat the next 50 cars that arrive as a simple random sample. There is a $15 \%$ chance the mean time will be at or below ____ seconds

Sheryl Ezze

Numerade Educator

Problem 27

The Food and Drug Administration sets Food Defect Action Levels (FDALs) for some of the various foreign substances that inevitably end up in the food we eat and liquids we drink. For example, the FDAL for insect filth in peanut butter is 3 insect fragments (larvae, eggs, body parts, and so on ) per 10 grams. A random sample of 50 ten-gram portions of peanut butter is obtained and results in a sample mean of $\bar{x}=3.6$ insect fragments per ten-gram portion.
(a) Why is the sampling distribution of $\bar{x}$ approximately normal?
(b) What is the mean and standard deviation of the sampling distribution of $\bar{x}$ assuming that $\mu=3$ and $\sigma=\sqrt{3} ?$
(c) What is the probability that a simple random sample of 50 ten-gram portions results in a mean of at least 3.6 insect fragments? Is this result unusual? What might we conclude?

Sneha Ravi

Numerade Educator

Problem 28

Suppose that cars arrive at Burger King's drive-through at the rate of 20 cars every hour between 12: 00 noon and 1: 00 P.M. A random sample of 40 one-hour time periods between 12: 00 noon and 1: 00 p.m. is selected and has 22.1 as the mean number of cars arriving.
(a) Why is the sampling distribution of $\bar{x}$ approximately normal?
(b) What is the mean and standard deviation of the sampling distribution of $\bar{x}$ assuming that $\mu=20$ and $\sigma=\sqrt{20} ?$
(c) What is the probability that a simple random sample of 40 one-hour time periods results in a mean of at least 22.1 cars? Is this result unusual? What might we conclude?

Sheryl Ezze

Numerade Educator

Problem 29

The amount of time Americans spend watching television is closely monitored by firms such as $\mathrm{AC}$ Nielsen because this helps determine advertising pricing for commercials.
(a) Do you think the variable "weekly time spent watching television" would be normally distributed? If not, what shape would you expect the variable to have?
(b) According to the American Time Use Survey, adult Americans spend 2.35 hours per day watching television on a weekday. Assume that the standard deviation for "time spent watching television on a weekday" is 1.93 hours. If a random sample of 40 adult Americans is obtained, describe the sampling distribution of $\bar{x},$ the mean amount of time spent watching television on a weekday.
(c) Determine the probability that a random sample of 40 adult Americans results in a mean time watching television on a weekday of between 2 and 3 hours.
(d) One consequence of the popularity of the Internet is that it is thought to reduce television watching. Suppose that a random sample of 35 individuals who consider themselves to be avid Internet users results in a mean time of 1.89 hours watching television on a weekday. Determine the likelihood of obtaining a sample mean of 1.89 hours or less from a population whose mean is presumed to be 2.35 hours. Based on the result obtained, do you think avid Internet users watch less television?

Sheryl Ezze

Numerade Educator

Problem 30

According to Crown ATM Network, the mean ATM withdrawal is \$67. Assume that the standard deviation for withdrawals is $\$ 35 .$
(a) Do you think the variable "ATM withdrawal" is normally distributed? If not, what shape would you expect the variable to have?
(b) If a random sample of 50 ATM withdrawals is obtained, describe the sampling distribution of $\bar{x},$ the mean withdrawal amount.
(c) Determine the probability of obtaining a sample mean withdrawal amount between $\$ 70$ and $\$ 75 .$

Sneha Ravi

Numerade Educator

Problem 31

The following data represent the ages of the winners of the Academy Award for Best Actor for the years $2004-2009$
$$
\begin{array}{lc}
\hline \text { 2004: Jamie Foxx } & 37 \\
\hline \text { 2005: Philip Seymour Hoffman } & 38 \\
\hline \text { 2006: Forest Whitaker } & 45 \\
\hline \text { 2007: Daniel Day-Lewis } & 50 \\
\hline \text { 2008: Sean Penn } & 48 \\
\hline \text { 2009: Jeff Bridges } & 60 \\
\hline
\end{array}
$$
(a) Compute the population mean, $\mu$.
(b) List all possible samples with size $n=2 .$ There should be ${ }_{6} C_{2}=15$ samples.
(c) Construct a sampling distribution for the mean by listing the sample means and their corresponding probabilities.
(d) Compute the mean of the sampling distribution.
(e) Compute the probability that the sample mean is within 3 years of the population mean age.
(f) Repeat parts (b)-(e) using samples of size $n=3$. Comment on the effect of increasing the sample size.

Shu Naito

Numerade Educator

Problem 32

The following data represent the running lengths (in minutes) of the winners of the Academy Award for Best Picture for the years $2004-2009$
$$
\begin{array}{lc}
\hline \text { 2004: } \text { Million Dollar Baby } & 132 \\
\hline \text { 2005: Crash } & 112 \\
\hline \text { 2006: The Departed } & 151 \\
\hline \text { 2007: No Country for Old Men } & 122 \\
\hline \text { 2008: Slumdog Millionaire } & 120 \\
\hline \text { 2009: The Hurt Locker } & 131 \\
\hline
\end{array}
$$
(a) Compute the population mean, $\mu$.
(b) List all possible samples with size $n=2 .$ There should be ${ }_{6} C_{2}=15$ samples.
(c) Construct a sampling distribution for the mean by listing the sample means and their corresponding probabilities.
(d) Compute the mean of the sampling distribution.
(e) Compute the probability that the sample mean is within 5 minutes of the population mean running time.
(f) Repeat parts (b)-(e) using samples of size $n=3$. Comment on the effect of increasing the sample size.

Rashmi Sinha

Numerade Educator

Problem 33

In the game of roulette, a wheel consists of 38 slots numbered $0,00,1,2, \ldots, 36$. (See the photo.) To play the game, a metal ball is spun around the wheel and is allowed to fall into one of the numbered slots. If the number of the slot the ball falls into matches the number you selected, you win $\$ 35$ otherwise you lose $\$ 1$
(a) Construct a probability distribution for the random variable $X,$ the winnings of each spin.
(b) Determine the mean and standard deviation of $f$ the random variable $X$ Round your results to the nearest penny.
(c) Suppose that you play the game 100 times so that $n=100$. Describe the sampling distribution of $\bar{x},$ the mean amount won per game.
(d) What is the probability of being ahead after playing the game 100 times? That is, what is the probability that the sample mean is greater than 0 for $n=100 ?$
(e) What is the probability of being ahead after playing the game 200 times?
(f) What is the probability of being ahead after playing the game 1000 times?
(g) Compare the results of parts (d) and (e). What lesson does this teach you?

Carson Merrill

Numerade Educator

Problem 34

Explain what a sampling distribution is.

Neel Faucher

Numerade Educator

Problem 35

State the Central Limit Theorem

Jon Southam

Numerade Educator

Problem 36

We assume that we are obtaining simple random samples from infinite populations when obtaining sampling distributions. If the size of the population is finite, we technically need a finite population correction factor. However, if the sample size is small relative to the size of the population, this factor can be ignored. Explain what an "infinite population" is. What is the finite population correction factor? How small must the sample size be relative to the size of the population so that we can ignore the factor? Finally, explain why the factor can be ignored for such samples.

Neel Faucher

Numerade Educator

Problem 37

Without doing any computation, decide which has a higher probability, assuming each sample is from a population that is normally distributed with $\mu=100$ and $\sigma=15 .$ Explain your reasoning.
(a) $P(90 \leq \bar{x} \leq 110)$ for a random sample of size $n=10$.
(b) $P(90 \leq \bar{x} \leq 110)$ for a random sample of size $n=20$.

Neel Faucher

Numerade Educator

Problem 38

For the three probability distributions shown, rank each distribution from lowest to highest in terms of the sample size required for the distribution of the sample mean to be approximately normally distributed. Justify your choice.

Neel Faucher

Numerade Educator

Problem 39

Suppose Jack and Diane are each attempting to use simulation to describe the sampling distribution from a population that is skewed left with mean 50 and standard deviation $10 .$ Jack obtains 1000 random samples of size $n=3$ from the population, finds the mean of the 1000 samples, draws a histogram of the means, finds the mean of the means, and determines the standard deviation of the means. Diane does the same simulation, but obtains 1000 random samples of size $n=30$ from the population.
(a) Describe the shape you expect for Jack's distribution of sample means. Describe the shape you expect for Diane's distribution of sample means.
(b) What do you expect the mean of Jack's distribution to be? What do you expect the mean of Diane's distribution to be?
(c) What do you expect the standard deviation of Jack's distribution to be? What do you expect the standard deviation of Diane's distribution to be?

Sneha Ravi

Numerade Educator

Problem 40

Suppose you want to study the number of hours of sleep you get each evening. To do so, you look at the calendar and randomly select 10 days out of the next 300 days and record the number of hours you sleep.
(a) Explain why number of hours of sleep in a night by you is a random variable.
(b) Is the random variable "number of hours of sleep in a night" quantitative or qualitative?
(c) After you obtain your ten nights of data, you compute the mean number of hours of sleep. Is this a statistic or a parameter? Why?
(d) Is the mean number of hours computed in part (c) a random variable? Why? If it is a random variable, what is the source of variation?

Sheryl Ezze

Numerade Educator

Problem 41

Suppose you want to study the number of hours of sleep full-time college students at your college get each evening. To do so, you obtain a list of full-time students at your college, obtain a simple random sample of ten students, and ask each of them to disclose how many hours of sleep they obtained the most recent Monday.
(a) What is the population of interest in this study? What is the sample?
(b) Explain why number of hours of sleep in this study is a random variable.
(c) After you obtain your ten observations, you compute the mean number of hours of sleep. Is this a statistic or a parameter? Why?
(d) Is the mean number of hours computed in part (c) a random variable? Why? If it is a random variable, what is the source of variation? How does the source of variation in this study differ from that of Problem $40 ?$

Sheryl Ezze

Numerade Educator

Problem 42

A bull market is defined as a market condition in which the price of a security rises for an extended period of time. A bull market in the stock market is often defined as a condition in which a market rises by $20 \%$ or more without a $20 \%$ decline. The data on the next page represent the number of months and percentage change in the $\mathrm{S} \& \mathrm{P} 500$ (a group of 500 stocks) during the 25 bull markets dating back to 1929 (the year of the famous market crash).
(a) Treating the length of the bull market as the explanatory variable, draw a scatter diagram of the data.
(b) Determine the linear correlation coefficient between months and percent change.
(c) Does a linear relation exist between duration of the bull market and market performance?
(d) Find the least-squares regression line treating length of the bull market as the explanatory variable.
(e) Interpret the slope.
(f) Did the bull market that lasted 50.4 months have a percent change above or below what would be expected? Explain.
(g) Draw a residual plot. Any outliers?
(h) Would you consider the bull market from December 4,1987 through March 24,2000 , which lasted 149.8 months and saw a $582.15 \%$ rise in stock prices, influential? Explain. Note:
After this bull market, the market entered a bear market that lasted 18.2 months and saw the stock market decline $37 \% .$ This era is often referred to as the "Tech Bubble."
$$
\begin{array}{cc|cc}
\begin{array}{l}
\text { Bull } \\
\text { Months }
\end{array} & \begin{array}{l}
\text { Percent } \\
\text { Change }
\end{array} & \begin{array}{l}
\text { Bull } \\
\text { Months }
\end{array} & \begin{array}{l}
\text { Percent } \\
\text { Change }
\end{array} \\
\hline 4.9 & 46.77 & 86.9 & 267.08 \\
\hline 2.3 & 25.83 & 50.4 & 86.35 \\
\hline 0.8 & 25.82 & 44.1 & 79.78 \\
\hline 1.2 & 30.61 & 26.1 & 48.05 \\
\hline 2.3 & 111.59 & 32.0 & 73.53 \\
\hline 4.7 & 120.61 & 74.9 & 125.63 \\
\hline 3.7 & 37.28 & 61.3 & 228.81 \\
\hline 24.2 & 131.64 & 149.8 & 582.15 \\
\hline 7.4 & 62.24 & 3.5 & 21.40 \\
\hline 6.6 & 26.78 & 60.9 & 101.50 \\
\hline 5.0 & 20.91 & 1.6 & 24.22 \\
\hline 49.7 & 157.70 & 48.6 & 127.85 \\
\hline 13.1 & 23.89 & & \\
\hline
\end{array}
$$

Shu Naito

Numerade Educator