Improve Your Understanding of College Level Statistics Problems

1) Describe sampling distributions and sampling variavility
2) Explain The Central Limit Theorem
3) Explain how confidence intervals are created and what can they tell us about population parameters

An automobile tire manufacturer collected the data in the table relating tire pressure x​ (in pounds per square​ inch) and mileage​ (in thousands of​ miles). A mathematical model for the data is given by
${\displaystyle f\left(x\right)=-0.554x^2+35.5x-514.}$
$\begin{array}{|cc|}\hline x& Mileage\\ 28& 45\\ 30& 51\\ 32& 56\\ 34& 50\\ 36& 46\\ \hline\end{array}$
​(A) Complete the table below.
$\begin{array}{|ccc|}\hline x& Mileage& f(x)\\ 28& 45\\ 30& 51\\ 32& 56\\ 34& 50\\ 36& 46\\ \hline\end{array}$
​(Round to one decimal place as​ needed.)
$A.20602060xf(x)$
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,45), (30,51), (32,56), (34,50), and (36,46). A parabola opens downward and passes through the points (28,45.7), (30,52.4), (32,54.7), (34,52.6), and (36,46.0). All points are approximate.
$B.20602060xf(x)$
Acoordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2.
Data points are plotted at (43,30), (45,36), (47,41), (49,35), and (51,31). A parabola opens downward and passes through the points (43,30.7), (45,37.4), (47,39.7), (49,37.6), and (51,31). All points are approximate.
$C.20602060xf(x)$
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (43,45), (45,51), (47,56), (49,50), and (51,46). A parabola opens downward and passes through the points (43,45.7), (45,52.4), (47,54.7), (49,52.6), and (51,46.0). All points are approximate.
$D.20602060xf(x)$
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,30), (30,36), (32,41), (34,35), and (36,31). A parabola opens downward and passes through the points (28,30.7), (30,37.4), (32,39.7), (34,37.6), and (36,31). All points are approximate.
​(C) Use the modeling function​ f(x) to estimate the mileage for a tire pressure of 29
${\displaystyle \frac{lbs}{sq}\in .}$ and for 35
${\displaystyle \frac{lbs}{sq}\in .}$
The mileage for the tire pressure ${\displaystyle 29\frac{lbs}{sq}\in .}$ is
The mileage for the tire pressure ${\displaystyle 35\frac{lbs}{sq}}$ in. is
(Round to two decimal places as​ needed.)
(D) Write a brief description of the relationship between tire pressure and mileage.
A. As tire pressure​ increases, mileage decreases to a minimum at a certain tire​ pressure, then begins to increase.
B. As tire pressure​ increases, mileage decreases.
C. As tire pressure​ increases, mileage increases to a maximum at a certain tire​ pressure, then begins to decrease.
D. As tire pressure​ increases, mileage increases.

The local energy company claims the average annual electricity bill for its subscribers is just $600. A consumer watchdog group wants to dispute this claim. All agree that the standard deviation sigma of annual electricity bills is $150.
Some time later, a wealthy activist provides funding for a simple random sample of 250 households. The average annual electricity bill for this sample is $622. Find a $95\mathrm{\%}$ confidence interval for the true mean annual electric bill, based on this sample.

Convert the point from spherical coordinates to rectangular coordinates.
$(9,\pi ,\frac{\pi }{2})$
$(x,y,z)=?$

Let's say the widget maker has developed the following table that shows the highest dollar price p. widget where you can sell N widgets. Number N Price p $20053.00$
$25052.50$
$30052.00$
$35051.50$

(a) Find a formula for pin terms of N modeling the data in the table.

(b) Use a formula to express the total monthly revenue R, in dollars, of this manufacturer in month as a function of the number N of widgets produced in a month. $R=$ Is Ra linear function of N?

(c) On the basis of the tables in this exercise and using cost, $C=35N+900$, use a formula to express the monthly profit P, in dollars, of this manufacturer asa function of the number of widgets produced in a month $p=$ ?

Let $f(x)=4-\frac{2}{x}+\frac{6}{x^2}$.
Find the open intervals on which f is increasing (decreasing). Then determine the x-coordinates of all relative maxima (minima).
f is increasing on the intervals
f is decreasing on the intervals
The relative maxima of f occur at x =
The relative minima of f occur at x =
Notes: In the first two, your answer should either be a single interval, such as (0,1), a comma separated list of intervals, such as (-inf, 2), (3,4), or the word "none".
In the last two, your answer should be a comma separated list of x values or the word "none".

Calculate confidence intervals for ratio of two population variances and ratio of standard deviations. Assume that samples are simple random samples and taken from normal populations. a) ${\displaystyle \alpha =0.05,n_1=30,s_1=16.37,n_2=39,s_2=9.88}$ b) ${\displaystyle \alpha =0.01,n_1=25,s_1=5.2,n_2=20,s_2=6.8}$

Which of the following is true about the sampling distribution of means?
A. Shape of the sampling distribution of means is always the same shape as the population distribution, no matter what the sample size is.
B. Sampling distributions of means are always nearly normal.
C. Sampling distributions of means get closer to normality as the sample size increases.
D. Sampling distribution of the mean is always right skewed since means cannot be smaller than 0.

In a study of the accuracy of fast food​ drive-through orders, Restaurant A had 298 accurate orders and 51 that were not accurate. a. Construct a 90​% confidence interval estimate of the percentage of orders that are not accurate. b. Compare the results from part​ (a) to this 90​% confidence interval for the percentage of orders that are not accurate at Restaurant​ B: ${\displaystyle 0.127<p<0.191}$. What do you​ conclude? a. Construct a 90​% confidence interval. Express the percentages in decimal form. ___

A distribution of values is normal with a mean of 203.8 and a standard deviation of 50.2.
Find the probability that a randomly selected value is less than 113.4.
${\displaystyle P\left(X<113.4\right)=}$
Write your answers as numbers accurate to 4 decimal places.

Decide equation using only factors to solve this problem
$12x^2+5x-2=0$

Define the term Mean.

Why is hair color classified as categorical data?

To Explain:the expected the mean, median,standard deviation and IQR to change.

The correct statement which is incorrect from the options about the sampling distribution of the sample mean
(a) the standard deviation of the sampling distribution will decrease as the sample size increases,
(b) the standard deviation of the sampling distribution is a measure of the variability of the sample mean among repeated samples,
(c) the sample mean is an unbiased estimator of the true population mean,
(d) the sampling distribution shows how the sample mean will vary in repeated samples,
(e) the sampling distributions shows how the sample was distributed around the sample mean.

Describe a possible application of hypothesis testing to a business setting. The application you describe need not be a real one (one that has actually occurred), it can be an application that you invent. You should describe a) what the business setting is b) what data you would use c) what null hypothesis you would test and what the alternative hypothesis is. d) what type of test you would use e) You should explain what level of significance you would use and why. f) You should explain how to interpret the outcome of the test: what does it tell you if you could reject or could not reject the null hypothesis? g) Finally, you should explain the possible advantages of using hypothesis testing in this application and its possible downsides dangers (when how using a hypothesis test could lead to mistaken inferences or erroneous decisions).

Decide which of the following statements are true. Answer: square Normal distributions are bell-shaped, but they do not have to be symmetric. square The line of symmetry for all normal distributions is x square On any normal distribution curve, you can find data values more than 5 standard v square deviations above the mean. square The x-axis is a horizontal asymptote for all normal distributions.

The homebuilder’s association reports that $75\mathrm{\%}$ of home buyers would like a fireplace in their new home. A local builder finds that 20 out of 30 of his customers wanted a fireplace. Using the p value decision rule determine at the 5% level of significance is there is enough evidence from the sample to conclude that the homebuilders report is too high?
If P is too high in ${\displaystyle H_0}$ then what should it be in ${\displaystyle H_1}$?
Also construct a ${\displaystyle (1-\alpha )\mathrm{\%}}$ interval estimate for the true proportion P.
Make sure to state the meaning of P first…and completely interpret the CI for P.

Explain the effect of correcting the error has on the IQR and the standard deviation.
The professor calculated the IQR and the standard deviation of the test scores in question 6 before realizing her mistake that is she entered the top score as 46 but it was actually 56.

It is estimated that aproximately ${\displaystyle 8.36\mathrm{\%}}$ Americans are afflicted with Diabetes .
Suppose that a ceratin diagnostic evaluation for diabetes will correctly diagnose ${\displaystyle 94.5\mathrm{\%}}$ of all adults over 40 with diabetes as having the disease and incorrectly diagnoses ${\displaystyle 2\mathrm{\%}}$ of all adults over 40 without diabetes as having the disease .
1) Find the probability that a randamly selected adult over 40 doesn't have diabetes and is diagnosed as having diabetes ( such diagnoses are called "false positives").
2) Find the probability that a randomly selected adult of 40 is diagnosed as not having diabetes.
3) Find the probability that a randomly selected adult over 40 actually has diabetes , given that he/she is diagnosed as not having diabetes (such diagnoses are called "false negatives").
Note: It will be helpful to first draw an appropriate tree diagram modeling the situation.