Exercises: Chi-square and regression inference

A market researcher wants to know if the distribution of car colors in Los Angeles matches the national distribution. Nationally, $25\%$ of cars are white, $20\%$ are black, $15\%$ are silver, $10\%$ are red, and $30\%$ are other colors. The researcher selects a random sample of $400$ cars in Los Angeles. Calculate the expected count for red cars to be used in a chi-square goodness-of-fit test.

A high school administrator is investigating if there is an association between a student's grade level (Freshman, Sophomore, Junior, Senior) and their preferred method of transportation to school (Bus, Car, Walk). The data is organized in a two-way table. The total number of Juniors surveyed is $120$, the total number of students who walk is $80$, and the grand total of all students surveyed is $500$. Calculate the expected count of Juniors who walk, assuming the null hypothesis of no association is true.

In a chi-square goodness-of-fit test comparing observed frequencies of a categorical variable to expected frequencies, one of the categories has an observed count of $45$ and an expected count of $50$. Calculate the individual contribution of this category to the overall chi-square test statistic.

A real estate agent runs a linear regression to predict the price of a house (in thousands of dollars) based on its size (in hundreds of square feet). The computer output is shown below:

Identify the value of the sample slope and the standard error of the slope from this output.

An ecologist is studying the relationship between the elevation (in feet) and the average daily temperature (in degrees Fahrenheit) in the Rocky Mountains. A least-squares regression line is fitted to a sample of $25$ locations. The estimated slope is $b = -0.004$ and the standard error of the slope is $SE_b = 0.0012$. Calculate the test statistic for testing the null hypothesis $H_0: \beta_1 = 0$.

A political scientist conducts a survey to determine if there is a difference in the distribution of political party affiliation (Democrat, Republican, Independent, Green) across three different regions of a state (Urban, Suburban, Rural). The results are tallied in a contingency table. Determine the degrees of freedom for the appropriate chi-square test.

Determine whether the following scenario requires a chi-square test for homogeneity or a chi-square test for independence: A researcher selects a single random sample of $300$ registered voters in Texas and asks each voter for their highest level of education (High School, College, Graduate) and their opinion on a new tax bill (Support, Oppose, No Opinion).

A consumer advocate is analyzing the relationship between the weight of a car (in pounds) and its fuel efficiency (in miles per gallon). Based on a random sample of $18$ cars, the slope of the least-squares regression line is $-0.006$ with a standard error of $0.0015$. The critical value for a $95\%$ confidence interval is $t^* = 2.120$. Construct the expression for the $95\%$ confidence interval for the true population slope.

The manufacturer of a popular candy claims that the color distribution of their product is $20\%$ red, $20\%$ orange, $20\%$ yellow, $20\%$ green, and $20\%$ purple. A student buys a large bag containing $150$ candies and counts $38$ red, $25$ orange, $22$ yellow, $35$ green, and $30$ purple candies. Calculate the chi-square test statistic to test the manufacturer's claim and state the degrees of freedom.

A university researcher wants to know if the type of extracurricular activity a student participates in (Sports, Arts, Academic Clubs) is independent of their academic major (STEM, Humanities, Business). A random sample of $240$ students yields a chi-square test statistic of $\chi^2 = 11.45$. Determine the degrees of freedom for this test and interpret the meaning of the P-value if it is found to be $0.022$.

A marine biologist investigates the relationship between the length of a certain species of shark (in inches) and its weight (in pounds). The computer output for the regression analysis of a sample of $22$ sharks is provided:

Calculate the $90\%$ confidence interval for the true slope of the regression line. Assume the conditions for inference are met. (Use $t^* = 1.725$ for $20$ degrees of freedom).

Two different manufacturing plants in Ohio produce the same electronic component. A quality control manager takes a random sample of $200$ components from Plant A and $300$ components from Plant B, categorizing them as Defective, Minor Flaw, or Perfect.

In Plant A, $10$ are Defective, $30$ have a Minor Flaw, and $160$ are Perfect.

In Plant B, $25$ are Defective, $45$ have a Minor Flaw, and $230$ are Perfect.

Identify the expected count for "Defective components from Plant A" and check if the expected count condition for the chi-square test of homogeneity is satisfied.

A fitness coach believes that the number of hours a person spends sleeping per night positively affects their maximum weight lifted (in pounds) on the bench press. A regression analysis on a random sample of $30$ athletes yields a slope of $b = 5.2$ and a standard error $SE_b = 2.4$. The computer output provides a two-sided P-value of $0.0385$. Assuming all conditions for inference are met, determine the P-value for the coach's specific hypothesis and state the conclusion at the $\alpha = 0.05$ significance level.

A data analyst is running a linear regression to predict the annual maintenance cost of delivery trucks based on their total mileage. The analyst checks the conditions for inference by examining the residual plot provided in Panel A. Based on the visual evidence, identify which specific condition for linear regression inference is clearly violated and explain why.

A sociologist conducts a chi-square test for independence to see if there is an association between a person's generation (Gen Z, Millennials, Gen X, Boomers) and their primary news source (Social Media, TV, Print, Radio). After calculating the expected counts, she notices that the expected count for "Boomers who use Social Media" is $3.2$ and for "Gen Z who use Print" is $2.5$. All other expected counts are greater than $5$. Describe the appropriate statistical remedy to satisfy the conditions for the chi-square test without collecting more data, and state the new degrees of freedom if this remedy is applied.

An AP Statistics student rolls a standard six-sided die $120$ times to test if it is fair. The observed frequencies for the faces $1$ through $6$ are $15, 25, 18, 22, 16,$ and $24$, respectively. Calculate the expected count for each face, compute the chi-square test statistic, and determine if there is convincing evidence at the $\alpha = 0.10$ level that the die is not fair.

A software company wants to know if the proportion of bugs reported in three different modules of their application matches the historical distribution. Historically, Module A accounts for $50\%$ of bugs, Module B for $30\%$, and Module C for $20\%$. A recent update was pushed, and out of a random sample of $200$ new bug reports, $90$ were for Module A. You are given that the total chi-square test statistic for this sample is $\chi^2 = 6.5$. Determine the exact number of observed bugs for Module B and Module C. (Assume the observed count for Module B is greater than the expected count for Module B).

An economist is studying the relationship between a city's unemployment rate ($x$, in percent) and its crime rate ($y$, in incidents per $1,000$ residents). A random sample of $45$ US cities is selected. The computer output is lost, but the economist remembers the following summary statistics: the sample correlation is $r = 0.65$, the standard deviation of the unemployment rates is $s_x = 2.4$, and the standard deviation of the crime rates is $s_y = 15.6$. Calculate the standard error of the slope ($SE_b$) using the formula $SE_b = \frac{s}{s_x \sqrt{n-1}}$, given that the standard deviation of the residuals is $s = 12.1$. Then, construct a $99\%$ confidence interval for the population slope.

A researcher is investigating the relationship between the dosage of a new medication ($X$) and the reduction in blood pressure ($Y$). The scatterplot in Panel 1 includes all data points, with a highly influential point labeled as Point P. Panel 2 shows the data with Point P removed.

Let $t_1$ and $p_1$ be the test statistic and P-value for testing $H_0: \beta_1 = 0$ in Panel 1. Let $t_2$ and $p_2$ be the corresponding values for Panel 2.

Analyze the visual data and explain theoretically how removing Point P will affect the standard error of the slope ($SE_b$), the magnitude of the test statistic ($t$), and the P-value. Will the evidence for a linear relationship become stronger or weaker?

In genetic crosses, the Hardy-Weinberg principle and Mendelian genetics predict that the offspring of a certain dihybrid cross should exhibit four distinct phenotypes in a $9:3:3:1$ ratio. A botanist performs this cross and observes $400$ plants. The observed counts are $245$ for the first phenotype, $65$ for the second, $70$ for the third, and $20$ for the fourth.

Carry out a complete chi-square goodness-of-fit test. If the botanist decides to combine the last two phenotypes into a single category due to their biological similarity, how does this modification change the degrees of freedom and the critical value of the test at the $\alpha = 0.05$ level?

A public health official is conducting a chi-square test for independence to determine if there is an association between a person's smoking status (Current, Former, Never) and their risk of developing a specific respiratory disease (High, Low). The test yields a P-value of $0.08$ at a significance level of $\alpha = 0.05$.

Define what a Type II error would mean in the context of this specific study. Furthermore, explain conceptually how increasing the sample size from $150$ to $1,500$ patients (assuming the sample proportions remain identical) would affect the expected counts, the chi-square test statistic, and the probability of making a Type II error.

A scatterplot of the price of a stock ($y$) against time in days ($x$) shows a clear exponential growth pattern. A data scientist applies a natural logarithm transformation to the $y$-variable and fits the least-squares regression line $\widehat{\ln(y)} = 1.2 + 0.045x$. The standard error of the slope is $0.012$ based on a sample of $30$ days.

Construct a $95\%$ confidence interval for the slope of the transformed regression line. Then, interpret the upper bound of this interval in terms of the original variables, specifically detailing the percentage growth of the stock price per day.

A school district wants to compare the pass rates of a standardized math test across four different high schools. A random sample of $50$ students is taken from each school. The district runs a chi-square test for homogeneity and finds a test statistic of $\chi^2 = 12.8$.

A separate analyst decides to pool the data from all four schools and run a single-sample proportion $z$-test comparing the overall pass rate to a hypothesized state average.

Discuss why the chi-square test is the more appropriate method for comparing the four schools directly, and mathematically demonstrate the relationship between the chi-square test statistic and the two-sample $z$-test statistic if the district had only compared two schools instead of four.

A sports analyst has the computer output for a linear regression predicting the number of wins a baseball team will have based on their run differential (Runs Scored minus Runs Allowed). The sample size is $30$ teams. The P-value for the two-sided hypothesis test $H_0: \beta_1 = 0$ is exactly $0.014$, and the estimated slope is $b = 0.105$.

The analyst accidentally deletes the standard error of the slope ($SE_b$) from the output. Using the given information, reconstruct the value of $SE_b$. (Hint: You will need to use an inverse t-distribution function or table to find the t-statistic corresponding to the P-value). Assume the test statistic is positive.