A company manufactures safety helmets and inspects $150$ helmets each week for defects. The number of defective helmets found in each of $10$ consecutive weeks is as follows: $5$, $8$, $6$, $4$, $7$, $9$, $11$, $13$, $10$, and $14$. Using a $p$-chart, determine whether the process is in statistical control. If not, explain why.

A population has p = 0.3. If a random sample of n = 40 is taken, what is the mean and standard deviation of the sampling distribution of p̂?

A population of size $N=80000$ has proportion $p=0.45$ with a characteristic. A simple random sample of size $n=600$ is taken. Approximate the probability that the sample contains at most $x=260$ individuals with the characteristic, using the normal approximation with finite population correction and a continuity correction.

What is the formula for the standard deviation of the sampling distribution of the sample proportion?

In a Gallup poll of $450$ employed adults, $82\%$ said they prefer working remotely at least part of the time. Among the $450$ respondents, how many said they prefer working remotely?

The durability times (in hours) of $12$ randomly selected ceiling fans have a sample standard deviation of $0.4$ hours. Assume the data come from a normally distributed population. Using an $80\%$ confidence level, construct the confidence interval for the population variance (${\sigma }^2$). Also, interpret the results.

A $2017$ survey found that $28\%$ of U.S. adults regularly shop for groceries online. Researchers now want to estimate the current percentage of adults who shop online for groceries. To obtain a $95\%$ confidence level and a margin of error of $2$ percentage points, how many adults must be surveyed now? Does the use of the $2017$ survey result significantly affect the required sample size?

A researcher reports a $99\%$ confidence interval for a population proportion. What does the "$99\%$ confidence" signify?

A medical study aims to estimate the proportion of patients responding to a new treatment. The researchers want a margin of error of $5\%$ at a $95\%$ confidence level. What is the minimum sample size required?

Which function on the TI-84 calculator is used to compute a confidence interval for a population proportion?

In a recent football cup, on‐field decisions were referred to the Video Assistant Referee (VAR) system $150$ times. Of those reviews, $40$ calls were overturned. Construct a $99\%$ confidence interval for the true proportion $p$ of VAR reviews that result in an overturned decision.

A software‐reliability engineer executes $80$ independent test runs of a new application and observes zero failures. She wants a $95\%$ confidence interval for the true failure probability $p$. Which statement best explains why the usual normal‐approximation confidence intervals fail when $x=0$, and why the “Rule of Three” provides a valid upper bound?

A recent poll surveyed $52,480$ adults about their exercise habits. Out of those surveyed, $18,412$ reported that they exercise at least $3$ times per week. Construct a $95\%$ confidence interval for the true proportion of adults who exercise at least $3$ times per week. Interpret the result.

A safety inspector wants to estimate, with $95\%$ confidence, the proportion of construction site accidents that occur due to equipment failure. The estimate must be accurate within $5\%$ of the actual proportion. Since there are no existing studies or historical data available about machinery-related accidents in this industry, what is the smallest number of accident reports the safety officer must examine to meet these precision requirements?

The weights (in kilograms) of $41$ randomly selected boxes of sugar had a sample standard deviation of $1.8\text{ kg}$. Assume the weights are normally distributed. Using a $98\%$ confidence level, construct the confidence interval for the population variance (${\sigma }^2$). Also, interpret the results.