Understanding Sampling Distribution with Plainmath

1.      Twenty plants werer randomly selected from a normal population of palm trees to measure their height. the sample mean of the height was found to be 5 m

a.      Check if there is any significant difference between the sample mean and a  population mean of 6 m with standard deviation of 2 m

b.      Below is a plant of residuals versus fits after a straight –line model was used on data for y= sale price of home and X = square foot area of home

A population of values has a normal distribution with μ=211.7${\displaystyle \mu =211.7}$ and σ=23.8${\displaystyle \sigma =23.8}$. You intend to draw a random sample of size n=232${\displaystyle n=232}$. Please answer the following questions, and show your answers to 1 decimal place.

Find the value separating the bottom 30% values from the top 70% values. 


Find the sample mean separating the bottom 30% sample means from the top 70% sample means.

P(X>2.8) =

The geyser Old Faithful in Yellowstone National Park alternates periods of eruption, which typically last from 1.5 to 4 minutes, with periods of dormancy, which are considerably longer. The following table presents the durations, in minutes,30 of dormancy periods that occurred during a recent year.Construct a frequency distribution using a class width of 7, and using 55 as the lower class limit for the first class.

For a distribution with mean 100 and standard deviation 20, a Z-score of 4.5 corresponds to a value

Find the area under the standard normal distribution by shading the direction of the following z- scores

 1. To the left of z=2.246

2. To the right of z=2.10

Between z=1.50 and z=2.25

1  1/2 + 1  1/2 =

Suppose that a random sample of 50 bottles of a particular brand of cough medicine is selected and the alcohol content of each bottle measured. The sample mean alcohol content is 8.6 ml and the standard deviation is 2.99 ml. The standard deviation of the alcohol content is required, by law, to be less than 2.8 ml.

A 90% confidence interval for the population variance of the alcohol content is

( 6.60 ; 12.91 ).

Thus, we can conclude with 90% confidence, that

A retail store has recently started paying attention to customer analytics. The store collected data from a sample of 139 walk-in customers about registration in the store's points program, purchase level, and payment method.

The amount of pollutants that are found in waterways near large cities is normally distributed with mean 8.6 ppm and standard deviation 1.4 ppm. 17 randomly selected large cities are studied. Round all answers to 4 decimal places where possible.

1. What is the distribution of X${\displaystyle X}$? X${\displaystyle X}$ ~ N(,)
2. What is the distribution of ¯x${\displaystyle \overline{x}}$? ¯x${\displaystyle \overline{x}}$ ~ N(,)
3. What is the probability that one randomly selected city's waterway will have more than 8.8 ppm pollutants? 
4. For the 17 cities, find the probability that the average amount of pollutants is more than 8.8 ppm. 
5. For part d), is the assumption that the distribution is normal necessary? YesNo
6. Find the IQR for the average of 17 cities.
   Q1 =  ppm
   Q3 =  ppm
   IQR:  ppm

According to the empirical rule, about 95% of the data lies within 2 standard deviations from the mean.

True or False?

art 1: Introduction (Please type this portion - approximately 1-2 pages)
1) Identify Your Topic of Study (These are some EXAMPLES that correspond to 4 different
types of questions; kindly think of your own!)
a) In this study we intend to investigate the proportion of men who believe that at
least one parent amongst two should stay at home with a child until he/she starts
going to school.
b) In this study we investigate the difference in proportion between men and women
who believe that at least one parent amongst two should stay at home with a
child until he/she starts going to school.
c) In this study we intend to investigate the average number hours teenagers spend
on their phones each day.
d) In this study we intend to investigate the difference in the average number of
hours teenagers spend on phones each day and the average number of hours
spent on phones by adults 20 years of age or above.
2) Why is your study important? See if you can find some articles,movies, scientific articles,
anything that addresses your topic. What do they say? It doesn’t have to actually
answer your question - just talk about the issue at least and why it may be significant in
your life, or in life in general.
3) Identify your population(s), and how you would IDEALLY go about collecting a good
sample for study. (In reality you’ll probably end up doing a convenience sample)
4) Make a hypothesis BASED ON YOUR DATA (you can explain why its not what you
expected, or why it is what you expected as well)(. You will want to make a hypothesis
that is about a PROPORTION, or about a MEAN. It can be for one population, or for 2
populations* For example:
a) My hypothesis is that the proportion will be more than 50% because....
b) My hypothesis is that more men think so than women because... (*2 population)
c) My hypothesis is that the average number of hours is more than 3 hours
d) My hypothesis is that the average number of hours teenagers in USA spend on
their phones daily is more than the average for adults in USA (*2 population)
Note: *The computations for 2 populations are slightly more complicated than for studies on a
single population - hence you will get extra credit for doing a paper on 2 populations (up to 10
points on top of the possible 100 points for the project)
At least 1 page, but not more than 2 pages double spaced!

Part 2: Data Collection and display (Please type up, use technology to make displays)
1) Collect your Sample Data. Collect data from at least 15-20 subjects. If your are
comparing 2 populations, then collect at least 7-8 subjects in each population.
2) Explain what you collected. What question(s) did you ask? How did you find the people
in your sample?
3) Show your data in some form: Below is the proportion of men who believe at least 1
parent should stay at home until a child is ready to go to school.
Similarly can display histograms or any other kind of display.

I want to us this as example :

Globally, the majority (57%) of consumers own pets, according to more than 27,000 online consumers whom GfK surveyed in 22 countries. Dogs are the most popular pet globally, owned by 33% of respondents, with cats coming in second, at 23%. Fish (12%), birds (6%), and other pet types (6%) all rank significantly lower.

Argentina (82%), Mexico (81%) and Brazil (76%) have the highest rates of overall pet ownership among the 22 countries, with dogs being the most popular pet in all three nations. The countries included are Argentina, Australia, Belgium, Brazil, Canada, China, Czech Republic, France, Germany, Hong Kong, Italy, Japan, Mexico, Netherlands, Poland, Russia, South Korea, Spain, Sweden, Turkey, UK and USA.

The US ranks fifth for pet ownership and third when it comes to owning cats. Almost three-quarters (70%) of US consumers have at least one pet, with 50% owning dogs and 39% cats.

The research shows significant similarities in pet ownership metrics among US men and women. Only 28% of men and 31% of women report not owning any pets. In results that echo the global averages, men and women report almost identical levels of ownership for dogs (52%, men; 48%, women), cats (40%, men; 38%, women) and fish (12%, men; 10% women).

Asian countries appear to have the smallest percentage of their online population who own pets. In South Korea, just 31 percent report having any pet living with them, followed by Hong Kong at 35 percent and Japan at 37 percent.

“Although the Asian countries come lower in this list, these countries comprise a significant and growing share of the global pet market,” said Pushan Tagore, vice president of pet care research at GfK in a press release. ”This is due to the overall size and growth rate of their human population… Looking at data from our full range of pet care research, the key markets to watch over the next 12 months are China, India and LATAM. In these markets, rising disposable income is moving consumers away from table scraps and home cooked food for dogs and cats and towards prepared pet food. This is being driven by the convenience factor, as well as rising awareness of the need to feed their pets with the appropriate nutrition.”

To gather this data, GfK conducted an online survey of over 27,000 consumers aged 15 or older in 22 countries. Fieldwork was completed in June 2015, and data are weighted to reflect the demographic composition of the online population age 15 and over in each market. 

A clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjects. Before treatment, 25 subjects had a mean wake time of 104.0 min. After treatment, the 25 subjects had a mean wake time of 94.7 min and a standard deviation of 22.4 min. Assume that the 25 sample values appear to be from a normally distributed population and construct a 99% confidence interval estimate of the mean wake time for a population with drug treatments

(a) Construct the 99% confidence interval estimate of the mean wake time for a population with the treatment.

(b) What does the result suggest about the mean wake time of 104.0 min before the treatment? 

(c) Does the drug appear to be effective?

 An underground military installation is fortified to the extent that it can withstand up to three 
direct hits from air-to-surface missiles and still function. Suppose an enemy aircraft is armed 
with missiles, each having a 30 % chance of scoring a direct hit. 
i. What is the probability that the installation will be destroyed with the seventh missile fired
is obtained by fewer than nine air strikes? 
ii. Find the expected air-to-surface missiles strike. 

A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 121.1-cm and a standard deviation of 2.2-cm. For shipment, 26 steel rods are bundled together.

Find P91, which is the average length separating the smallest 91% bundles from the largest 9% bundles.
P91 = -cm

Enter your answer as a number accurate to 2 decimal place. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.