Improve Your Understanding of College Level Statistics Problems

A researcher obtains a negative value for the chi-square statistic in a chi-square goodness of fit test. What can a researcher conclude based on this result?
choices:
The observed frequencies are consistently larger than the expected frequencies.
The expected frequencies are consistently larger than the observed frequencies.
There are large differences between the observed and expected frequencies.
The researcher made a mistake because a chi-square value cannot be negative.

A researcher is interested in assessing if there are differences in the average number of monthly religious event attendances across religious affiliations. Religious event attendance is measured as an interval-ratio variable: it is the number of events that each survey respondent reported attending in the previous month. Religious affiliation is measured as a nominal variable with the following categories: Protestant, Catholic, Jewish, Muslim, Hindu, Buddhist, other, and not religious. Which of the following hypothesis tests would be appropriate, given the types of variables presented?
a. Chi-square test
b. Two-sample difference-of-proportions z-test
c. ANOVA
d. One-sample difference-of-proportions z-test

A random sample of 175 students was selected from a large high school. They were each asked their current grade level and whether they had a driver's license. The responses are displayed in the table.
$\begin{array}{cc}& \text{Grade Level}\\ \text{Driver's License?}& \begin{array}{|ccccc|}\hline & \text{ 9th }& \text{ 10th }& \text{ 11th }& \text{ 12th}\\ \text{ Yes }& 1& 5& 21& 48\\ \text{ No }& 28& 42& 20& 10\\ \hline\end{array}\end{array}$
The principal would like to know if these data provide convincing evidence of a relationship between grade level and having a driver's license in the population of all students in her school. What is the appropriate inference procedure?
1. chi-square test for goodness of fit because the data come from one random sample
2. chi-square test for homogeneity because the data come from independent random samples
3. chi-square test for association/independence because the data come from one random sample
4. chi-square test for association/independence because the data come from independent random samples

A board game in a fieste carnival has the numbers 2 through 12. These numbers are the sum of the results in rolling todice. If you are going to place your bet on the numbers, what will you choose and why? Apply your knowledge on probabilitydistribution Show your solutions

A college lecturer randomly selected thirty-five students in his class to complete an assigned task andit is found that the 95% confidence interval for the true average time in completing the task is(17.278, 28.722) minutes. It is thought that this interval is too wide, and a confidence interval of totalwidth 5 minutes will be more desirable.(a) Using the same value of , find the new confidence interval in this case.(b) Find the confidence level for the interval in part (a).

A researcher obtains a Pearson correlation of r = 0.60 for a sample of n = 6 pairs of X and Y scores. If the researcher tests the significance of the correlation, what value will be obtained for the t statistic?

A physician who has a group of thirty-eight female patients aged 18 to 24 on a special diet wishes to estimate the effect of the diet on total serum cholesterol. For this group, their average serum cholesterol is 188.4 (measured in mg/100mL). Because of a large-scale government study, the physician is willing to assume that the total serum cholesterol measurements are normally distributed with standard deviation of ${\displaystyle \sigma =40.7}$. Find a 95% confidence interval of the mean serum cholesterol of patients on the special diet. Does the diet seem to have any effect on their serum cholesterol, given that the national average for women aged 18 to 24 is 192.0?

A manufacturing company with 350 employees is changing the employee health insurance plan to either plan A or plan B. The company wants to know if employees have a preference between the two plans and whether or not preference differs between those employees who have family members covered under the current plan (group 1) and those who do not (group 2). The human resources office takes a simple random sample from each of the two groups, sends information about both plans to the employees in each sample, and asks them whether they prefer plan A or plan B. The table summarizes the responses received, with expected cell counts in parentheses.
$\begin{array}{|cccc|}\hline & \text{ Plan A }& \text{ Plan B }& \text{ Total }\\ \text{ Yes (group 1) }& 40(32.5)& 20(27.5)& 60\\ \text{ No (group 2) }& 6(13.5)& 19(11.5)& 25\\ \text{ Total }& 46& 39& 85\\ \hline\end{array}$
Which statement is true about whether the conditions for the chi-square test for homogeneity have been met?
a. A simple random sample should have been taken from all the employees, and then each employee in the sample should have been asked their plan preference and whether or not they have family members covered under the current health insurance plan.
b. The expected cell counts are not large enough to apply the chi-square test for homogeneity.
c. The total sample size is not large enough to apply the chi-square test for homogeneity.
d. The total sample size is too large to apply the chi-square test for homogeneity.
e. All conditions necessary to apply the chi-square test for homogeneity are satisfied here.

A local pizza store knows the mean amount of time it takes them to deliver an order is 45 minutes arter the order is placed. The manager has a new system for processing delivery orders, and they want to test ifit changes the mean delivery time. They take a sample of delivery orders and find their mean delivery time is 48 minutes.
Let ${\displaystyle \mu }$ represent the mean delivery time under the new system.
What are appropriate hypotheses for their signifhcance test?
a)${\displaystyle H_0:\mu }$= 45 minutes
${\displaystyle H_a:\mu }$ ${\displaystyle \ne }$45 minutes
b)${\displaystyle H_0:\mu }$ =45 minutes
${\displaystyle H_a:\mu }$ ${\displaystyle \ne }$48 minutes
c)${\displaystyle H_0:\mu }$ p= 48 minutes
${\displaystyle H_a:\mu }$ ${\displaystyle \ne }$45 minutes
d)${\displaystyle H_0:\mu }$ = 48 minutes
${\displaystyle H_a:\mu }$ ${\displaystyle \ne }$48 minutes

A job candidate at a large job fair can be classified as unacceptable, provisional, or acceptable. Based on past experience, a high-quality candidate is expected to get 80 percent acceptable ratings, 15 percent provisional ratings, and 5 percent unacceptable ratings. A high-quality candidate was evaluated by 100 companies and received 60 acceptable, 25 provisional, and 15 unacceptable ratings. A chi-square goodness-of-fit-test was conducted to investigate whether the evaluation of the candidate is consistent with past experience. What is the value of the chi-square test statistic and number of degrees of freedom for the test?
a. ${\displaystyle {\chi }^2=\frac{{(15-5)}^2}{5}+\frac{{(25-15)}^2}{15}+\frac{{(60-80)}^2}{80}}$ with 2df
b. ${\displaystyle {\chi }^2=\frac{{(15-5)}^2}{5}+\frac{{(25-15)}^2}{15}+\frac{{(60-80)}^2}{80}}$ with 3df
c. ${\displaystyle {\chi }^2=\frac{{(15-5)}^2}{5}+\frac{{(25-15)}^2}{15}+\frac{{(60-80)}^2}{80}}$ with 99df
d. ${\displaystyle {\chi }^2=\frac{{(5-15)}^2}{15}+\frac{{(15-25)}^2}{25}+\frac{{(80-60)}^2}{60}}$ with 2df
e. ${\displaystyle {\chi }^2=\frac{{(5-15)}^2}{15}+\frac{{(15-25)}^2}{25}+\frac{{(80-60)}^2}{60}}$ with 3df

A controversial issue in the sport of professional soccer is the use of instant replay for making difficult goal line decisions. Each person in a representative sample of 102 players, fans, coaches, and officials was asked his or her opinion about the use of instant replay for goal-line decisions. The data are summarized in the two-way frequency table below.
$\begin{array}{cc}& \text{Opinion}\\ \text{Category}& \begin{array}{|ccc|}\hline & \text{ Favor Use }& \text{ Oppose Use }\\ \text{ Players }& 22& 2\\ \text{ Fans }& 18& 6\\ \text{ Coaches }& 15& 26\\ \text{ Officials }& 3& 10\\ \hline\end{array}\end{array}$
In testing to see whether opinion with respect to the use of instant replay is independent of the category of the person interviewed, a chi-square test statistic of 27.99 and a p-value less than 0.001 were calculated. Which of the following statements is correct?
a. The number of degrees of freedom for the test is 8 - 1 = 7.
b. The chi-square test should not have been used because two of the counts in the table are less than 5.
c. The null hypothesis states that there is an association between category and opinion about the use of instant replay, and the small p-value suggests that the null hypothesis should be rejected.
d. The small p-value suggests that there is evidence of an association between category and opinion about the use of instant replay.
e. The chi-square test shows that fans favor the use of instant replay.

A clinical trial is run to investigate the effectiveness of an experimental drug in reducing preterm delivery to a drug considered standard care and to placebo. Pregnant women are enrolled and randomly assigned to receive either the experimental drug, the standard drug or placebo. Women are followed through delivery and classified as delivering preterm (< 37 weeks) or not. The data are shown below. Is there a statistically significant difference in the proportions of women delivering preterm among the three treatment groups? Run the test at a 5% level of significance. Give each of the following
1) the appropriate null and alternative hypotheses; 2) the appropriate test; 3) the decision rule; 4) the calculation of the test statistic; and 5) your conclusion including a comparison to alpha or the critical value.
$\begin{array}{cc}& \text{Preterm Delivery}\\ & \begin{array}{|cccc|}\hline & \text{ Yes }& \text{ No }& \text{ Total }\\ \text{ Experimental Drug }& 7& 33& 40\\ \text{ Standard Drug }& 13& 32& 45\\ \text{ Placebo }& 17& 18& 35\\ \text{ Total }& 37& 83& 120\\ \hline\end{array}\end{array}$

A clarification on the level of confidence for a Chi-squared distribution
Suppose we have ${\displaystyle {\chi }_{1-\alpha ,29}^2=45.22}$, where the Chi-squared distribution has 29 degrees of freedom and ${\displaystyle (1-\alpha )\mathrm{\%}}$ level of confidence. How then do we compute ${\displaystyle 1-\alpha }$? The notation of ${\displaystyle {\chi }^2}$. I've found in books and online sources all come in the form of ${\displaystyle {\chi }_c^2}$, where c is the degrees of freedom.
I know this question seems trivial, but I'm still relatively new to Statistics.. Some clarification will be great!

1. Which of the following accurately describes the observed and expected frequencies in a chi-square test?
a. Both the observed and expected frequencies will always be whole numbers.
b. The observed frequencies are always whole numbers but the expected frequencies may be decimals or fractions.
c. The expected frequencies are always whole numbers but the observed frequencies may be decimals or fractions.
d. Both the observed and expected frequencies may be decimals or fractions.
2. What does it mean to obtain a negative value for the chi-square statistic?
a. The observed frequencies tend to be larger than the expected frequencies.
b. The expected frequencies tend to be larger than the observed frequencies.
c. There are large discrepancies between the observed and expected frequencies for most categories.
d. The chi-square statistic can never be negative.

"Determine the critical values for these tests of a population standard deviation.
(a) A right-tailed test with 18 degrees of freedom at the ${\displaystyle \alpha =0.01}$ level of significance
(b) A left-tailed test for a sample of size n= 24 at the ${\displaystyle \alpha =0.1}$ level of significance
(c) A two-tailed test for a sample of size n= 26 at the ${\displaystyle \alpha =0.01}$ level of significance"
(a) The critical value for this right-tailed test is $◻$ (Round to three decimal places as needed.)
(b) The critical value for this left-tailed test is $◻$ ( Round to three decimal places as needed.)
(c) The critical values for this two-tailed test are $◻$ (Round to three decimal places as needed. Use a comma to separate answers as needed.)

Suppose that the height, in inches, of a 20-year-old man is a normal random variable
with parameters μ = 175 and σ = 10.

There are ten 20-year-old men in the class. What's the probability that more than 8
of them are over 185cm?

 

90% Confidence Interval for Difference Between Two Proportions
In random, independent samples of 225 adults and 250 teenagers who watched a certain television show, 99 adults and 125 teens indicated that they liked the show. Let ${\displaystyle p_1}$ be the proportion of all adults watching the show who liked it, and let ${\displaystyle p_2}$ be the proportion of all teens watching the show who liked it. Find a 90% confidence interval for ${\displaystyle p_1-p_2}$.
1. what is the lower limit of the 90% confidence interval
2. what is the upper limit of the 90% confidence interval

1) In using chi-square for testing the independence of variables, larger calculated values for chi-square will tend to make it more likely that the null hypothesis will be rejected.
True or False?
a) A contingency table for two nominal variables has three (3) row and four (4) columns. The number of degrees of freedom associated with the chi-square test of variable independence will be df= 6.
True or False?
b) In carrying out chi-square analysis, if one or more of the observed frequencies is less than five (5), rows or columns should be combined so that each value of observed frequency is at least 5.
True or False?

1. When the null hypothesis for a chi square test for independence is true.
a. a large difference between the observed frequencies and the expected frequencies.
b.little or no difference between the observed frequencies and the expected frequencies.
c. no difference between the observed frequencies and the marginals
d. no differences between the row and the column marginal
2. In the Chi Square test, degrees of freedom are calculated as :
a. N-1
b. ${\displaystyle N_1+N_2-2}$
c. (r+1)(c+1)
d. (r-1)(c-1)

1. For each of the following test z- test versus the t-test versus the chi squared test. Describe what the test is for and what conditions must be satisfied before test is conducted
2. Give at least one example of two variables that have positive correlation. Discuss why we would not be able to deduce causation in your example