Data Modeling Examples and Frequently Asked Questions

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.

Gastroenterology
We present data relating protein concentration to pancreatic function as measured by trypsin secretion among patients with cystic fibrosis.
If we do not want to assume normality for these distributions, then what statistical procedure can be used to compare the three groups?
Perform the test mentioned in Problem 12.42 and report a p-value. How do your results compare with a parametric analysis of the data?
Relationship between protein concentration $(mg/mL)$ of duodenal secretions to pancreatic function as measured by trypsin secretion:
$[U/\frac{kg}{hr}]$
Tapsin secreton [UGA]
$\le 50$
$\begin{array}{|cc|}\hline \text{Subject number}& \text{Protetion concentration}\\ 1& 1.7\\ 2& 2.0\\ 3& 2.0\\ 4& 2.2\\ 5& 4.0\\ 6& 4.0\\ 7& 5.0\\ 8& 6.7\\ 9& 7.8\\ \hline\end{array}$
$51-1000$
$\begin{array}{|cc|}\hline \text{Subject number}& \text{Protetion concentration}\\ 1& 1.4\\ 2& 2.4\\ 3& 2.4\\ 4& 3.3\\ 5& 4.4\\ 6& 4.7\\ 7& 6.7\\ 8& 7.9\\ 9& 9.5\\ 10& 11.7\\ \hline\end{array}$
$>1000$
$\begin{array}{|cc|}\hline \text{Subject number}& \text{Protetion concentration}\\ 1& 2.9\\ 2& 3.8\\ 3& 4.4\\ 4& 4.7\\ 5& 5.5\\ 6& 5.6\\ 7& 7.4\\ 8& 9.4\\ 9& 10.3\\ \hline\end{array}$

Identifying Probability Distributions. In Exercises 7–14, determine whether a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied. Cell Phone Use In a survey, cell phone users were asked which ear they use to hear their cell phone, and the table is based on their responses (based on data from “Hemispheric Dominance and Cell Phone Use,” by Seidman et al., JAMA Otolaryngology—Head & Neck Surgery , Vol. 139, No.

5). $\overline{)\begin{array}{cc} & P\left(x\right)\\ Left& 0.636\\ Right& 0.304\\ No preference& 0.060\end{array}}$

An analysis of laboratory data collected with the goal of modeling the weight (in grams) of a bacterial culture after several hours of growth produced the least squares regression line $\log (weight)=0.25+0.61$hours. Estimate the weight of the culture after 3 hours.

A) 0.32 g

B) 2.08 g

C) 8.0 g

D) 67.9 g

E) 120.2 g

According to the article “Modeling and Predicting the Effects of Submerged Arc Weldment Process Parameters on Weldment Characteristics and Shape Profiles” (J. of Engr. Manuf., 2012: 1230–1240), the submerged arc welding (SAW) process is commonly used for joining thick plates and pipes. The heat affected zone (HAZ), a band created within the base metal during welding, was of particular interest to the investigators. Here are observations on depth (mm) of the HAZ both when the current setting was high and when it was lower. $\begin{array}{ccccccccccccccccccc}Non-high& 1.04& 1.15& 1.23& 1.69& 1.92& 1.98& 2.36& 2.49& 2.72& 1.37& 1.43& 1.57& 1.71& 1.94& 2.06& 2.55& 2.64& 2.82\\ High& 1.55& 2.02& 2.02& 2.05& 2.35& 2.57& 2.93& 2.94& 2.97& & & & & & & & & \\ & & & & & & & & & & & & & & & & & & \end{array}$ c. Does it appear that true average HAZ depth is larger for the higher current condition than for the lower condition? Carry out a test of appropriate hypotheses using a significance level of .01.

The article “Modeling Sediment and Water Column Interactions for Hydrophobic Pollutants” (Water Research, 1984: 1169-1174) assumes a uniform distribution over the interval (7.5, 20) as a model for depth (cm) of the bioturbation layer in sediment in a certain region. a. What are the average and spread of depth? b. What is the cdf of depth? c. What is the probability that observed depth is at most 10? Between 10 and 15? d. What is the probability that the observed depth is within 1 standard deviation of the mean value? Within 2 standard deviations?