Introduction to Confidence Intervals: Video e Problemi di Pratica

A research team wants to estimate the standard deviation of time spent on social media by teenagers. Find the minimum sample size needed to be $99\%$ confident that the sample standard deviation $s$ is within $3\%$ of the population standard deviation $\sigma$.

A circular access port is designed with a diameter of $20$ inches. The waist widths of workers are normally distributed with a mean of $17$ inches and a standard deviation of $0.9$ inches. What percentage of workers can pass through the access port?

A normal quantile plot is constructed for the test scores of $35$ students in a statistics class. The plot shows a pronounced $S$-shape, with the points curving upward at both ends and dipping in the middle. What does this indicate about the distribution of the test scores?

A sample of 20 students' scores on a statistics final exam are $62,65,67,68,70,71,72,73,74,75,76,77,78,79,80,81,82,83,85,88$. Does this sample suggest the scores are from a population that is approximately normal?

A sample of $25$ exam scores from a statistics class is as follows: $72,75,78,80,81,83,85,86,87,88,89,89,90,91,92,93,94,95,96,97,98,99,100,100,100$. Based on the distribution of these scores, does the data appear to be from a population with a normal distribution?

A sociologist is constructing a confidence interval for the average number of hours worked per week using a sample of $70$ individuals from a population with a left-skewed distribution. The population standard deviation is not provided. Which distribution should be used?

A public health researcher wants to estimate the average daily sugar intake of adults in the United States. She collects data only from adults living in Florida. Is the mean sugar intake from this sample a good estimate for the national average?

A group of $50$ volunteers participated in a mindfulness program. After $8$ weeks, their mean reduction in stress score was $4.6$ points, with a sample standard deviation of $5.3$ points. Construct a $99\%$ confidence interval estimate of the standard deviation of the reduction in stress scores for all such participants.

$12$ batteries were randomly selected and tested for their lifespans (in $\text{hours}$), producing the following values: $42.1,43.7,41.5,44.2,42.8,43.3,41.9,44.0,42.5,43.1,42.3,43.5$. The normal quantile plot of this data appears to follow a straight line. Are the necessary assumptions satisfied to construct a $95\%$ confidence interval for the population standard deviation? If so, calculate the interval.

Suppose you are conducting a two-sample t-test and have two sample sizes: $n_1=10$ and $n_2=15$. Using the conservative approach, you use degrees of freedom equal to the minimum of $n_1-1$ and $n_2-1$, which is $9$. This gives a critical value of $t=2.262$ for a one-tailed test at the $0.05$ significance level. If you instead use the formula for degrees of freedom and get $17.8$, the critical value is $t=1.740$. Why is using $t=2.262$ considered more conservative than $t=1.740$?

Listed below is a simple random sample of the number of times caffeine (e.g., coffee, energy drinks) was mentioned during episodes of a teen drama series.

$4,5,6,3,7,5,4,6,8,5$

Are the conditions met for constructing a $95\%$ confidence interval estimate for the population mean number of caffeine mentions per episode? If yes, compute the $95\%$ confidence interval for the mean.

A simple random sample of hours per week that top amateur marathon runners train is shown below.

$12,15,14,11,13,16,14,12,13,15$

Construct a $95\%$ confidence interval estimate of the population mean weekly training hours.

Researchers measured the simple reaction times (milliseconds) of a random sample of drivers and cyclists under identical conditions. From these samples, they computed a $95\%$ confidence interval for the difference in population mean reaction times (drivers minus cyclists) as:

What does this confidence interval suggest about whether the mean reaction time of drivers equals the mean reaction time of cyclists?

A researcher is studying the variation in weekly exercise hours among college students. They want to determine the sample size needed to estimate the population standard deviation $\sigma$, with $90\%$ confidence that the sample standard deviation $s$ is within $10\%$ of $\sigma$. Find the required sample size.

A quality control engineer collected a simple random sample of 10 bolts produced in a batch to assess their lengths (in millimeters). The sample has the following statistics:

Sample mean $\left(\bar{x}\right)=24.07\text{ mm}$

Margin of error (ME) for a $90\%$ confidence interval $=0.11\text{ mm}$

Construct the $90\%$ confidence interval for the true mean length of bolts produced in the first batch using the given statistics. Another confidence interval for a sample of bolts from a second batch is reported as $(23.95\text{ mm},24.25\text{ mm})$. How does the result compare to the confidence interval of the second batch? Explain possible reasons for differences in their widths.

A food testing agency is evaluating the amount of iron (in milligrams) per serving in breakfast cereals. Two independent random samples were tested.

Sample A $(n=15)$ produced a $95\%$ confidence interval for the mean iron content, equal to $(8.45,9.88)\text{ mg}$, while sample B $(n=12)$ had a $99\%$ confidence interval for the mean iron content of $(7.62,10.14)\text{ mg}$.

Given that the data are not from a normally distributed population, which of the two confidence intervals is likely to provide a better estimate of the population mean?

A health researcher collected a random sample of $30$ nightly sleep durations (in hours) from a fitness tracking device. To estimate the population median sleep time, the researcher generated $1000$ bootstrap samples, each of size $30$, by sampling with replacement from the original data. The median sleep time was computed for each bootstrap sample. After sorting the $1000$ bootstrap medians, the researcher found the following percentile values:

${2.5}^{th}$ percentile: $6.25$ hours

${97.5}^{th}$ percentile: $8.15$ hours

The median of all $500$ nightly sleep durations stored in the tracker’s database is known to be $7.1$ hours. How well did the bootstrap method perform in capturing a good estimate of the population median?

A team of nutrition researchers wants to estimate the proportion of grocery stores that stock low-sodium canned vegetables. Determine the minimum sample size required to estimate this proportion, using a margin of error of $0.10$ and a confidence level of $95\%$. Past surveys suggest that approximately $40\%$ of grocery stores carry such products.

A team evaluating sustainable packaging wants to estimate the proportion of retail brands that use biodegradable materials in their products. Find the minimum sample size required for this estimate, assuming a $95\%$ confidence level and a margin of error of $0.10$. Earlier research indicates that around $70\%$ of retail brands have adopted biodegradable packaging.

A transportation analyst wants to estimate the average delivery time (in days) for express shipping by a national courier service. Assume the population standard deviation is $\sigma =1.3$ days. How many shipments should be sampled to be $95\%$ confident that the estimate is within $0.2$ day of the true population mean?

A random sample of $8$ dimes minted after $2010$ has the following weights in grams: $2.27,2.29,2.28,2.30,2.26,2.28,2.29,2.27$. Construct a $95$ percent confidence interval for the mean weight of all such dimes. The Mint requires a standard deviation of $0.0200$ grams for dimes. What does your interval suggest about the Mint's process?

A normal quantile plot was created using the annual incomes (in thousands of dollars) of a random sample of $40$ teachers. What can be concluded about the distribution of the teachers' incomes?

A biologist reports three confidence intervals for the average height of a plant species, all calculated from the same sample and using the same method. The intervals correspond to 80%, 90%, and 95% confidence levels, but they are listed in random order. Which interval is most likely associated with the 90% confidence level?

True or False. When the sample size is $25$ and the population standard deviation is unknown, it is appropriate to use $z$-scores to determine the probability that the sample mean falls within a specified interval.

A researcher collects a random sample of $100$ students' test scores from a large university. The population of test scores is known to be right-skewed. What can be said about the shape of the sampling distribution of the sample mean?

True or False. The standard deviation of the distribution of sample means decreases as the number of observations in each sample increases.

Which of the following statements about the sampling distribution of the sample mean is correct?

A population has a mean of $845$ and a standard deviation of $10$. If a random sample of size $400$ is taken, what are the mean and standard deviation of the sampling distribution of the sample means?

Suppose a population has a mean of $3200$ and a standard deviation of $8$. If random samples of size $1600$ are selected, what are the mean and standard deviation of the sampling distribution of the sample means?

A certain population has a mean of $150$ and a standard deviation of $4$. If samples of size $64$ are drawn, what are the mean and standard deviation of the sampling distribution of the sample means?