Improve Your Understanding of College Level Statistics Problems

Give a complete answer and describe Bayes' theorem

A chi-square value of 6.15 is calculated from data in a $4x5$ contingency table. Assuming $\alpha =0.05$, identify the critical value.
26.217
31.410
21.026
16.919

A standard ${\displaystyle 3\sigma x-\stackrel{―}{}}$ chart has been enhanced with early warning limits at plus-minus one ${\displaystyle \sigma }$ from the centerline . Three sample means in a row have plotted above the ${\displaystyle +1\sigma }$ line . what is the probability of this happening if the process is still in control.

Let C be a circle, and let P be a point not on the circle. Prove that the maximum and minimum distances from P to a point X on C occur when the line X P goes through the center of C. [Hint: Choose coordinate systems so that C is defined by
$x2+y2=r2$ and P is a point (a,0)
on the x-axis with a $\ne \pm r,$ use calculus to find the maximum and minimum for the square of the distance. Don’t forget to pay attention to endpoints and places where a derivative might not exist.]

For a population with a mean of ${\displaystyle \mu =100}$ and a standard deviation of ${\displaystyle \sigma =20}$,
Find the X values.
${\displaystyle z=+1.80}$.

Census data are often used to obtain probability distributions for various random variables. Census data for families in a particular state with a combined income of $50,000 or more show that 22% of these families have no children, 30% have one child, 27% have two children, and 21% have three children. From this information, construct the probability distribution for x, where x represents the number of children per family for this income group. (Give your answers correct to two decimal places.) $P(0\text{children})=$
$P(1\text{child})=$
$P(2\text{children})=$
$P(3\text{children})=$

Use the Composite Trapezoidal, Simpson's and Midpoint Rules to approximate the integral
${\int }_1^{2}x\ln (x)dx,n=4$
(For the Midpoint Rule use $n+2$ sub intervals.)

Ozone The Environmental Protection Agency is examin- ing the relationship between the ozone level (in parts per million) and the population (in millions) of U.S. cities.
Part of the regression analysis is shown.
Dependent variable is: Ozone
R-squared $=84.4\mathrm{\%}$
$s=5.454$ with $16-2=14$ df
Variable Coefficient SE(Coeff)
Intercept 18.892 2.395
Pop 6.650 1.910
a) We suspect that the greater the population of a city, the higher its ozone level. Is the relationship significant? Assuming the conditions for inference are satisfied, test an appropriate hypothesis and state your conclu- sion in context.
b) Do you think that the population of a city is a useful predictor of ozone level? Use the values of both R2 and s in your explanation.

Check whether the standard error of the sampling distributions of $\bar{p}$ obtained in part(a) and part(b) are different.

A machine produces parts with lengths that are normally distributed with ${\displaystyle \sigma =0.66}$. A sample of 20 parts has a mean length of 75.98.
a)
Give a point estimate for ${\displaystyle \mu }$. (Give your answer correct to two decimal places.)
b)
Find the $95\mathrm{\%}$ confidence maximum error of estimate for ${\displaystyle \mu }$. (Give your answer correct to two decimal places.)
c)
Find the $95\mathrm{\%}$ confidence interval for ${\displaystyle \mu }$. (Give your answer correct to two decimal places.)
Lower limit ${\displaystyle =?}$
Upper limit ${\displaystyle =?}$

A random sample of 88 U.S. 11th- and 12th-graders was selected. The two-way table summarizes the gender of the students and their response to the question "Do you have allergies?" Suppose we choose a student from this group at random.

$\begin{array}{cccc}& \text{ Female }& \text{ Male }& \text{ Total }\\ \text{ Yes }& 19& 15& 34\\ \text{ No }& 24& 30& 54\\ \text{ Total }& 43& 45& 88\end{array}$
What is the probability that the student is female or has allergies?
$(a)\frac{19}{88}$
(b)$\frac{39}{88}$
(c)$\frac{58}{88}$
(d)$\frac{77}{88}$

You want to know whether people in different regions of the country are equally likely to vote Sarah Duterte, Peter Cayetano, Mar Roxas, or any candidate other than the three in the next election. You would use
A. chi-square test of independence.
B. either chi-square test (goodness-of-fit or test of independence), depending on how you set up the problem.
C. chi-square goodness-of-fit test.
D. both chi-square tests, in order to check the results of one with the other.

For a population with a mean of ${\displaystyle \mu =100}$ and a standard deviation of ${\displaystyle \sigma =20}$,
Find the X values.
${\displaystyle z=-.40}$.

Would you rather spend more federal taxes on art? Of a random sample of $n_1=86$ politically conservative voters, $r_1=18$ responded yes. Another random sample of $n_2=85$ politically moderate voters showed that $r_2=21$ responded yes. Does this information indicate that the population proportion of conservative voters inclined to spend more federal tax money on funding the arts is less than the proportion of moderate voters so inclined? Use $\alpha =0.05.$

a) State the null and alternate hypotheses.

 

$H_0:p_1=p_2,H_1:p_1>p_2$

$H_0:p_1=p_2,H_1:p_1<p_2$

$H_0:p_1=p_2,H_1:p_1\ne p_2$

$H_0:p_1<p_2,H_1:p_1=p_2$

b) What sampling distribution will you use? What assumptions are you making? The Student's t. The number of trials is sufficiently large. The standard normal. The number of trials is sufficiently large.The standard normal. We assume the population distributions are approximately normal. The Student's t. We assume the population distributions are approximately normal.

c)What is the value of the sample test statistic? (Test the difference $p_1-p_2$. Do not use rounded values. Round your final answer to two decimal places.)

d) Find (or estimate) the P-value. (Round your answer to four decimal places.)

e) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level alpha? At the $\alpha =0.05$ level, we reject the null hypothesis and conclude the data are statistically significant. At the $\alpha =0.05$ level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the $\alpha =0.05$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant. At the $\alpha =0.05$ level, we reject the null hypothesis and conclude the data are not statistically significant.

f) Interpret your conclusion in the context of the application. Reject the null hypothesis, there is sufficient evidence that the proportion of conservative voters favoring more tax dollars for the arts is less than the proportion of moderate voters. Fail to reject the null hypothesis, there is sufficient evidence that the proportion of conservative voters favoring more tax dollars for the arts is less than the proportion of moderate voters. Fail to reject the null hypothesis, there is insufficient evidence that the proportion of conservative voters favoring more tax dollars for the arts is less than the proportion of moderate voters. Reject the null hypothesis, there is insufficient evidence that the proportion of conservative voters favoring more tax dollars for the arts is less than the proportion of moderate voters.

Understanding the Concepts and Skills
Suppose that, for a sample of pairs of observations from two variables, the linear correlation coefficient, r , is negative. Does this result necessarily imply that the variables are negatively linearly correlated? Explain.

Make a scatterplot of the data with two new points added.
a) ${\displaystyle 10\mathrm{\%}}$ return, 25 new birds.
b) ${\displaystyle 40\mathrm{\%}}$ return, 5 new birds.
Find two new correlations: for the original data plus Point A and for the original data plus Point B.

Hypothesis Testing Review
For each problem below, simply identify the null and alternative hypotheses. Use appropriate notation/symbols. You do not have to run any hypothesis tests, although its

Discuss the importance of data modeling.

A population of values has a normal distribution with ${\displaystyle \mu =13.7}$ and ${\displaystyle \sigma =22}$.
You intend to draw a random sample of size ${\displaystyle n=78}$.
Find the probability that a single randomly selected value is less than 11.5.
${\displaystyle P\left(X<11.5\right)=}$?
Write your answers as numbers accurate to 4 decimal places.

What is nominal (categorial) data?