Exercises: Confidence intervals

A recent survey of $500$ randomly selected registered voters in a large county in Texas found that $280$ of them support a new property tax initiative. Calculate the point estimate for the true proportion of voters who support the initiative, and compute the margin of error for a $95\%$ confidence interval. Use $z^* = 1.96$.

The Department of Transportation in Georgia computes a $90\%$ confidence interval for the true mean commute time (in minutes) of residents in Atlanta. The calculated interval is $(28.4, 32.6)$. Identify the point estimate (sample mean) and the margin of error used to construct this interval.

A marine biologist is studying the average weight of a specific species of salmon in the Pacific Northwest. She constructs a $99\%$ confidence interval for the true mean weight and writes: "We are $99\%$ confident that the interval from $12.4$ pounds to $15.2$ pounds captures the true mean weight of this salmon species."

Explain exactly what the phrase "$99\%$ confident" means in the context of repeated sampling.

A quality control engineer at a factory in Michigan takes a random sample of $18$ car batteries and measures their lifespan in months. The sample mean is $42.5$ months with a sample standard deviation of $3.2$ months. Assuming the population of battery lifespans is approximately normally distributed, determine the critical value $t^*$ that should be used to construct a $95\%$ confidence interval for the true mean lifespan.

A fast-food chain wants to estimate the proportion of its customers who would purchase a new plant-based burger. They want the margin of error to be no more than $0.04$ with $95\%$ confidence ($z^* = 1.96$). Since they have no prior information about the true proportion, they use a conservative estimate of $\hat{p} = 0.5$. Calculate the minimum sample size required to meet these conditions.

To estimate the true mean price of a gallon of regular unleaded gas in Florida, a consumer advocacy group takes a simple random sample of $45$ gas stations. The sample mean is $\$3.45$ and the sample standard deviation is $\$0.18$. Calculate the $90\%$ confidence interval for the true mean price of a gallon of gas. Use $t^* = 1.680$.

A high school principal in Chicago wants to estimate the proportion of seniors who plan to attend a four-year college. She selects a simple random sample of $30$ seniors from the senior class of $800$ students. In the sample, $24$ students say they plan to attend a four-year college. Check if the "Large Counts" (Normality) condition is met for constructing a confidence interval for the population proportion. Show your work and state your conclusion.

A local bakery in New York City sells an average of $\bar{x} = 210$ bagels per day based on a random sample of $35$ days, with a margin of error of $14$ bagels for a $95\%$ confidence interval. Write out the $95\%$ confidence interval and provide a correct interpretation of the interval in the context of the problem.

A software company in Silicon Valley is testing a new user interface. They take a simple random sample of $150$ users and find that $96$ of them prefer the new interface over the old one.

Verify that all conditions for inference are met (assume the company has over $10,000$ users). Then, construct and interpret a $95\%$ confidence interval for the true proportion of all users who prefer the new interface.

An urban planner in Seattle is researching the monthly rent for one-bedroom apartments in the downtown area. She wants to estimate the true mean rent with $95\%$ confidence and a margin of error of no more than $\$50$. Based on previous studies, she assumes the population standard deviation is approximately $\sigma = \$220$.

Calculate the minimum sample size of apartments she needs to randomly select to achieve this margin of error.

A researcher is analyzing the wingspan of the American Bald Eagle. A random sample of $22$ adult eagles yields a mean wingspan of $79.2$ inches with a standard deviation of $3.8$ inches. A dotplot of the sample data shows no strong skewness and no outliers.

Construct a $98\%$ confidence interval for the true mean wingspan of adult American Bald Eagles. State the degrees of freedom and the critical value used.

A statistics student uses the same sample data to construct three different confidence intervals for a population mean, as shown in the visual above. The confidence levels used are $90\%$, $95\%$, and $99\%$.

Match each interval (A, B, and C) to its corresponding confidence level. Explain the mathematical relationship between the confidence level and the margin of error that justifies your answer, assuming the sample size and sample standard deviation remain constant.

A political campaign manager wants to conduct a poll to estimate the proportion of voters in Ohio who support their candidate. In a similar poll conducted two months ago, the support was estimated at $42\%$. Using this prior estimate, how many registered voters should the campaign randomly sample to estimate the true proportion within a margin of error of $3\%$ with $99\%$ confidence?

A local gym in Denver wants to estimate the average amount of time its members spend working out per visit. The manager stands by the front door on a Monday morning from 8:00 AM to 10:00 AM and asks the first $40$ members who leave how long they worked out. The $95\%$ confidence interval calculated from this data is $(45.2, 58.6)$ minutes.

Identify which condition for constructing a confidence interval is violated in this scenario. Explain how this violation could introduce bias into the estimate of the true mean workout time for all gym members.

An agricultural researcher calculates a $90\%$ confidence interval for the mean yield (in bushels per acre) of a new variety of corn in Iowa. Based on a simple random sample of $36$ plots, the interval is $(185.4, 194.6)$.

Using the formula for the margin of error of a one-sample t-interval, work backwards to determine the sample standard deviation ($s$) of the corn yields. (Use $t^* = 1.690$ for your calculation).

A public health official claims that more than half of the adults in a Texas county have received the annual flu vaccine. A random sample of $300$ adults in the county reveals that $162$ of them have received the vaccine.

Construct a $95\%$ confidence interval for the true proportion of adults in the county who have received the vaccine. Based strictly on your confidence interval, is there convincing evidence to support the official's claim? Explain your reasoning.

A popular coffee shop chain in Boston claims that their drip coffee is served at an optimal mean temperature of $160^\circ\text{F}$. A skeptical customer takes a random sample of $12$ cups of coffee over a month and records their temperatures. The dotplot of the sample data is shown above. The sample mean is $160.5^\circ\text{F}$ and the sample standard deviation is $4.6^\circ\text{F}$.

First, discuss whether the conditions for constructing a one-sample t-interval are satisfied, referencing the provided dotplot. If the conditions are met, calculate a $95\%$ confidence interval for the true mean temperature of the coffee. Does the interval provide convincing evidence that the coffee shop's claim is false?

An environmental agency wants to estimate the true proportion of households in California that have installed solar panels. They want the margin of error to be no more than $0.02$ with $99\%$ confidence.

The agency has a strict budget of $\$15,000$ for the survey, and each household sampled costs $\$4.50$ to process.

Determine the required sample size to meet the statistical goals (assuming no prior estimate for $p$). Can the agency achieve their desired margin of error and confidence level within their budget? If not, what is the smallest margin of error they can achieve with $99\%$ confidence using their entire budget?

A factory in Pennsylvania manufactures baseballs. The production manager takes a random sample of $400$ baseballs from the assembly line and finds that $12$ of them have defective stitching.

The manager wants to construct a confidence interval, but the intern points out a potential issue with the conditions for inference. Check the conditions for constructing a $95\%$ confidence interval for the true proportion of defective baseballs. If a condition is violated, identify it and explain why the standard $1$-Prop Z-Interval formula might not yield a strictly valid interval in this specific mathematical context.

A university researcher is studying the average student loan debt of recent graduates. She calculated a $95\%$ confidence interval for the mean debt based on a sample of $n = 50$ students. Her margin of error was $\$3,600$.

She decides this margin of error is too large and wants to reduce it to $\$1,200$ while maintaining the same $95\%$ confidence level. Assuming the sample standard deviation remains constant, algebraically demonstrate how many total students she needs to sample. Explain the general mathematical relationship between the sample size multiplier and the margin of error divisor.

A major trucking company in Texas calculates a $90\%$ confidence interval for the mean lifespan (in miles) of a specific brand of commercial tires. The interval is $(78,500, 82,100)$.

A manager reads the report and makes the following four statements:

1. "$90\%$ of the tires in our sample lasted between $78,500$ and $82,100$ miles."

2. "There is a $90\%$ probability that the true mean lifespan of all tires of this brand is between $78,500$ and $82,100$ miles."

3. "If we took $100$ different random samples of the same size, exactly $90$ of the resulting intervals would contain the true mean lifespan."

4. "We are $90\%$ confident that the sample mean lifespan is between $78,500$ and $82,100$ miles."

Identify why each of these four statements is a statistically incorrect interpretation of the confidence interval or the confidence level.

A researcher takes a random sample of $n = 5$ observations to estimate a population mean $\mu$. The sample mean is $\bar{x} = 104$ and the sample standard deviation is $s = 12$. The population is known to be normally distributed.

Initially, the researcher calculates a $95\%$ confidence interval using the appropriate t-distribution (shown as the dashed orange curve in the SVG). Suddenly, a previous census report is discovered, revealing that the true population standard deviation is exactly $\sigma = 12$. The researcher recalculates the $95\%$ confidence interval, this time using the standard normal distribution (solid blue curve).

Calculate both confidence intervals. Using the visual properties of the curves provided in the SVG, explain theoretically why the margin of error changes the way it does when moving from the t-distribution to the standard normal distribution.

A local school district in Illinois wants to estimate the true proportion of elementary school students who are reading below grade level. They take a simple random sample of $80$ students and find that $6$ of them are reading below grade level.

1. Show mathematically that the Large Counts condition for constructing a standard $1$-Prop Z-Interval is not met.

2. Because the condition fails, the standard interval is invalid. A statistician suggests using the "Plus Four" (Agresti-Coull) method, which adds $2$ successes and $2$ failures to the sample before calculating the interval. Calculate the new pseudo-sample proportion $\tilde{p}$ and construct the $95\%$ confidence interval using this adjusted method.

The city council of Austin, Texas, is considering a new municipal bond to fund public parks. To proceed, they need two pieces of information from their constituents:

1. The proportion of residents who support the bond.

2. The mean amount of additional annual property tax residents are willing to pay for these parks.

A random sample of $250$ residents is surveyed. Of those, $145$ support the bond. The sample mean for the amount they are willing to pay is $\$42.50$ with a sample standard deviation of $\$18.20$.

Construct a $99\%$ confidence interval for the true proportion of supporters, and a $95\%$ confidence interval for the true mean amount residents are willing to pay.

The council will only put the bond on the ballot if they are highly confident that a majority (more than $50\%$) support it, AND if the lower bound of the mean willingness to pay is at least $\$40.00$. Based on your intervals, what should the city council decide? Justify your answer.