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Exam 14: Goodness-Of-Fit Tests and Categorical Data Analysis

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A statistics department at a state university maintains a tutoring service for students in its introductory service courses. The service has been staffed with the expectation that 40% of its students would be from the business statistics course, 30% from engineering statistics, 20% from the statistics course for social science students, and the other 10% from the course for agriculture students. A random sample of n=120 students revealed 50, 40, 18, and 12 from the four courses. Does this data suggest that the percentages on which staffing was based are not correct? State and test the relevant hypotheses using $\alpha = .05$

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Using the number 1 for business, 2 for e...

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Criminologists have long debated whether there is a relationship between weather conditions and the incidence of violent crime. A study classified 1400 homicides according to season, resulting in the accompanying data. Test the null hypothesis of equal proportions using $\alpha = .01$ by using the chi-squared table to say as much as possible about the P-value. $\begin{array} { c c c c } \hline \text { Winter } & \text { Spring } & \text { Summer } & \text { Fall } \\\hline 338 & 345 & 382 & 335 \\\hline\end{array}$

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We test \[H _ { 0 } : p _ { 1 } = p _ { 2 } = p _ { 3 } = p _ { 4 } = 25 \text { vs } H _ { \pm } :\] at least one proportion \[\neq 25\] . We will reject \[H _ { 0 }\] if the p-value <.01. \[\begin{array} { c | c c c c } \hline \text { Cell } & 1 & 2 & 3 & 4 \\ \hline \text { Observed } & 338 & 345 & 382 & 335 \\ \left( n _ { i } \right) & & & & \\ \text { Expected } & 350 & 350 & 350 & 350 \\ \left( n _ { i } p _ { i } \right) & & & & \\ \hline & .411 & .071 & 2.926 & .643 \\ \chi ^ { 2 } & & & & \\ \text { term } & & & & \\ \hline \end{array}\] \[\chi ^ { 2 } = \sum \frac { ( \text { observed - expected } ) ^ { 2 } } { \text { expected } }\] = .411 + .071 + 2.926 +.643 = 4.051, and with 3 d.f., p-value >.10, so we fail to reject \(H _ { 0 } .\) The data fails to indicate a seasonal relationship with incidence of violent crime.

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Let $\hat { \theta } _ { 1 } , \ldots \ldots , \hat { \theta } _ { m }$ be the maximum likelihood estimators of the unknown parameters $\theta _ { 1 } , \ldots \ldots , \theta _ { m }$ , and let $\chi ^ { 2 }$ denote the test statistic value based on these estimators. If the data are classified into k categories, then the critical value $c _ { \alpha }$ that specifies a level $\alpha$ upper-tailed test satisfies

A study reports on research into the effect of different injection treatments on the frequencies of audiogenic seizures. Does the data suggest that the true percentages in the different response categories depend on the nature of the injection treatment? State and test the appropriate hypotheses using $\alpha = .005$

A study reports data on the rate of oxygenation in streams at $20 ^ { \circ }$ C in certain region. The sample mean and standard deviation were computed as $\bar { x } = .173 \text { and } s = .066 \text {, }$ respectively. Based on the accompanying frequency distribution, can it be concluded that oxygenation rate is a normally distributed variable? Use the chi-squared test with $\alpha = .05$ $\begin{array} { c c } \hline \text { Rate (per day) } & \text { Frequency } \\\hline \text { Below .100 } & 14 \\\hline .100 \text {-below } & 24 \\.150 & \\\hline .150 \text { - below } & 28 \\.200 & \\\hline .200 \text {-below } & 18 \\.250 & \\\hline .250 \text { or more } & 16 \\\hline\end{array}$

The chi-squared distribution has a single parameter $v$ , called the number __________ of the distribution.

A certain type of flashlight is sold with the four batteries included. A random sample of 150 flashlights is obtained, and the number of defective batteries in each is determined, resulting in the following data? $\begin{array} { c | c c c c c } \hline \text { Number of defective } & 0 & 1 & 2 & 3 & 4 \\\hline \text { Frequency } & 26 & 51 & 47 & 16 & 10 \\\hline\end{array}$ Let X be the number of defective batteries in a randomly selected flashlight. Test the null hypothesis that the distribution of X is Bin $( 4 , \theta )$ That is, with $p _ {i } = P ( i \text { defectives } )$ test $H _ { 0 } : p _ { i } = \left( \begin{array} { l } 4 \\i\end{array} \right) \theta^{t} ( 1 - \theta ) ^ { 4 - i }$ i=0,1,2,3,4 [Hint: To obtain the MLE of $\theta$ write the likelihood (the function to be maximized) as $\theta ^ { u } ( 1 - \theta ) ^ { \nu }$ where the exponents $u \text { and } v$ are linear functions of the cell counts. Then take the natural log, differentiate with respect to $\theta$ equate the result to 0, and solve for $\hat { \theta } .$ ]

A __________ generalizes a binomial experiment by allowing each trial to result in one of k possible outcomes (categories), where k > 2.

Which of the following statements are not true?

What conclusion would be appropriate for an upper-tailed chi-squared test in each of the following situations? a. $\alpha = .05 , \text { df } = 4 , { \chi } ^ { 2 } = 14.69$ b. $\alpha = .01 , d f = 3 , \chi ^ { 2 } = 7.85$ c. $\alpha = .10 , d f = 2 , { \chi } ^ { 2 } = 4.42$ d. $\alpha = 01 , k = 6 , \chi ^ { 2 } = 13.92$

The $\chi ^ { 2 }$ goodness-of-fit test statistics, when there are 6 categories and 2 parameters to be estimated, has approximately a chi-squared distribution with __________ degrees of freedom.

Which of the following statements are true regarding the critical value $\tilde { \chi } _ { \alpha , \nu }$ for the chi-squared distribution when $\alpha = .05 \text { and } v = 4 ?$

Each individual in a random sample of high school and college students was cross-classified with respect to both political views and marijuana usage, resulting in the data displayed in the accompanying two-way table. Does the data support the hypothesis that political views and marijuana usage level are independent within the population? Test the appropriate hypotheses using level of significance .01. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$ $\quad$$\text { Usage Level }$ $\begin{array}{l}\begin{array} { l | c c c } \hline \begin{array} { l } \text { Political } \\\text { Views }\end{array} & \text { Never } & \text { Rarely } & \text { Frequently } \\\hline \text { Liberal } & 480 & 180 & 120 \\\text { Conservative } & 215 & 50 & 15 \\\hline \text { Other } & 170 & 45 & 85 \\\hline\end{array}\end{array}$

The critical value $\hat { \chi } _ {.05,v } ^ { 2 }$ for the chi-squared distribution is the value such that __________ of the area under the $\chi ^ { 2 }$ curve with $v$ degrees of freedom lies to the right of $\chi _ { .05 , v } ^ { 2 }$

Consider the accompanying 2 $\times$ 3 table displaying the sample proportions that fell in the various combinations of categories (e.g., 13% of those in the sample were in the first category of both factors). $\begin{array} { l c c c } \hline & \mathbf { 1 } & \mathbf { 2 } & \mathbf { 3 } \\\hline \mathbf { 1 } & .13 & .19 & .28 \\\hline \mathbf { 2 } & .07 & .11 & .22 \\\hline\end{array}$ a. Suppose the sample consisted of n = 100 people. Use the chi-squared test for independence with significance level .10. b. Repeat part (a) assuming that the sample size was n = 1000. c. What is the smallest sample size n for which these observed proportions would result in rejection of the independence hypothesis?

Which of the following statements are not true?

The chi-squared test for homogeneity can safely be applied as long as each estimated expected county $\hat { e } _ { y }$ for all cells in the contingency table must be

Provided that $n p _ { i } \geq 5$ for every i (i =1, 2, 3, 4, 5), the $\chi ^ { 2 }$ goodness-of-fit test statistic when category probabilities are completely specified has approximately a chi-squared distribution with __________ degrees of freedom.

One may wish to test $H _ { 0 } : p _ { 1 } = \theta ^ { 2 } , p _ { 2 } = 2 \theta ( 1 - \theta ) , p _ { 3 } = ( 1 - \theta ) ^ { 2 } \text { versus } H _ {\pm } : H _ { 0 }$ is not true. The null hypothesis is__________ hypothesis because knowing that $H _ { 0 }$ is true does not uniquely determine the cell probabilities and expected cell counts but only their general form.

If $Z \square N ( 0,1 )$ , then $Z ^ { 2 }$ has a