Sampling Distribution of Means and Proportions

.8469

A camp accrediting association has 2,400 camps in their association. They claim that 45% of their camps offer specialized programs for individuals with disabilities. Suppose you contact a simple random sample of 200 of the camps and ask them whether they offer specialized programs for individuals with disabilities. Assuming that the association's claim is correct, what is the approximate probability that between 40% and 50% of the camps reply "yes" to your question?

.3085

A car manufacturer crash tests a certain model of car and measures the impact force. The test and model in question produce impact forces that are normally distributed with a mean of 30 metric tons and a standard deviation of 3 metric tons. Suppose that the manufacturer tests a random sample of 4 cars and calculates the sample mean impact force. What is the probability that the mean impact force from a sample of 4 cars x ̅ exceeds 30.75 metric tons?

.0668

A company sells eggs whose individual weights are normally distributed with a mean of 70 g and a standard deviation of 2 g. Suppose that these eggs are sold in packages that each contain 4 eggs that represent an SRS from the population. What is the probability that the mean weight of 4 eggs in a package xˉ is less than 68.5 g?

.0524

A consulting agency reports that only 30% of a company's website users can successfully use a certain feature of the website. Skeptical of this claim, one of the website developers takes a simple random sample of 150 of the company's approximately 8000 users and tests whether they can use the feature successfully. The developer finds that 36% of the sampled users use the feature successfully. Assuming the agency's 30% claim is correct, what is the approximate probability that more than 36% of the sample would use the feature successfully?

.0854

A human resources manager keeps a record of how many years each employee at a large company has been working in their current role. The distribution of these years of experience are strongly skewed to the right with a mean of 3 years and a standard deviation of 2 years. Suppose we were to take a random sample of 30 employees and calculate the sample mean for their years of experience. We can assume independence between members in the sample. What is the probability that the mean years of experience from the sample of 4 employees xˉ is greater than 3.5 years?

.8176

A large sleep study involving over 5,000 American teenagers examined how many hours the participants slept on the weekend. The nightly sleep times were distinctly non-normal with a mean of 10 hours and a standard deviation of 3 hours. Suppose we take a random sample of 100 nightly sleep times from this population. We can assume that the times in the sample are independent. What is the probability that the mean of these 100 sleep times xˉ is farther than 0.4 hours away from the population mean?

.0423

A local agricultural cooperative claims that 55% of about 60,000 adults in a county believe that gardening should be part of the school curriculum. However, when you take a simple random sample of 300 of the adults in the county, only 50% say that they believe that gardening should be part of the school curriculum. Assuming that the agricultural cooperative's claim is accurate, what is the approximate probability that less than 50% of the sample would say that they believe that gardening should be part of the school curriculum?

.8513

A manufacturer makes integrated circuits that each have a resistance layer with a target thickness of 200 units. A circuit won't work well if this thickness varies too much from the target value. These thickness measurements are approximately normally distributed with a mean of 200 units and a standard deviation of 12 units. A random sample of 16 measurements is selected for a quality inspection. We can assume that the measurements in the sample are independent. What is the probability that the mean thickness in these 16 measurements xˉ is farther than 3 units away from the target value?

.9199

A pizza chain monitors the total weight of pepperoni that goes on its deluxe pepperoni pizzas to make sure customers are satisfied and product isn't being wasted. Suppose that for pizzas in this population, these weights are strongly skewed to the left with a mean of 250 g and a standard deviation of 8 g. Management takes a random sample of 64 of these pizzas and calculates the mean weight of the pepperoni on the pizzas. Assume that the pizzas in the sample are independent. What is the probability that the mean weight of the pepperoni from the sample of 64 pizzas xˉ is within 1.75 g of the true mean?

Can't calculate because the population distribution is skewed to the right and the sample size is too small.

A pizza chain records how long it takes customers to receive their delivery orders. Suppose the distribution of these delivery times is strongly skewed to the right with a mean of 30 minutes and a standard deviation of 10 minutes. Management plans on calculating the mean delivery time from a random sample of 25 orders. We can assume independence between orders in the sample. What is the probability that the mean delivery time from the sample of 25 orders xˉ is farther than 2 minutes from the population mean?

.1587

A scientist studying water quality measures the lead level in parts per billion (ppb) at each of 49 randomly chosen locations along a water line. Suppose that the lead levels across all the locations on this line are strongly skewed to the right with a mean of 17 ppb and a standard deviation of 14 ppb. Assume that the measurements in the sample are independent. What is the probability that the mean lead level from the sample of 49 measurements xˉ is less than 15 ppb?

.0572

According to a survey, 63% of the Scottish population has visited woodland in the previous year. Skeptical about this claim, you decide to take a simple random sample of 650 people in this population, and you ask them if they had visited woodland in the previous year. You find that 60% of the sample replied "yes" to your question. Assuming that the original survey's 63% claim is correct, what is the approximate probability that less than 60% of the sample would report that they had visited woodland in the previous year?

.4215

According to the 2011 National Survey of Fishing, Hunting, and Wildlife-Associated Recreation, there were over 71 million wildlife watchers in the US. Of these wildlife watchers, the survey reports that 80% actively observed mammals. Suppose that one of the census workers repeated the survey with a simple random sample of only 500 wildlife watchers that same year. Assuming that the original survey's 80% claim is correct, what is the approximate probability that between 79% and 81% of the 500 sampled wildlife watchers actively observed mammals in 2011?

.0478

An article claimed that only 38% of hotel visitors used the alarm clock provided. A hotel manager wanted to see whether that proportion applied to the hotel where she worked, so she took a random sample of 700 of the approximately 10,000 visitors to the hotel one month and checked whether they used their room alarm clock. Of the sampled visitors, only 35% used the alarm clock. Assuming that the article's 38% claim is also the true proportion for visitors to the hotel, what is the approximate probability that less than 35% of the sample used the alarm clock?

.8944

An article written for a magazine claims that 78% of the magazine's subscribers report eating healthily the previous day. Suppose we select a simple random sample of 675 of the magazine's approximately 50,000 subscribers to check the accuracy of this claim. Assuming the article's 78% claim is correct, what is the approximate probability that less than 80% of the sample would report eating healthily the previous day?

.0228

Beatriz takes a simple random sample of 350 orders from the more than 5000 total orders to her store and finds that 12% of the sampled orders were returned. Assuming that it is really 9% of all orders to her store which were returned, what is the approximate probability that more than 12% of the sampled orders would have been returned?

.1319

Houseflies have pretty short lifespans. Males of a certain species have lifespans that are strongly skewed to the right with a mean of 26 days and a standard deviation of 12 days. A biologist collects a random sample of 45 of these male houseflies and observes them to calculate the sample mean lifespan. What is the probability that the mean lifespan from the sample of 45 houseflies xˉ is less than 24 days?

.9545

Individual bottles of water are filled by a machine at a factory with an amount of water that is approximately normal with a mean of 505 mL and a standard deviation of 10 mL. A random sample of 16 bottles is selected for a quality inspection. What is the probability that the mean amount of water in these 16 bottles xˉ is within 5 mL of the population mean?

.0092

Male mosquitos have pretty short lifespans. Males of a certain species have lifespans that are strongly skewed to the right with a mean of 8 days and a standard deviation of 6 days. A biologist collects a random sample of 50 of these male mosquitos and observes them to calculate the sample mean lifespan. What is the probability that the mean lifespan from the sample of 36 mosquitos xˉ exceeds 10 days?

Can't calculate because the population distribution is skewed to the left and the sample size is too small.

Mathieu grows specialty tomatoes that are much larger than typical tomatoes. The distribution of their weights is strongly skewed to the left with a mean of 232 g and a standard deviation of 12 g. Suppose we were to calculate the mean weight from a random sample of 16 of Mathieu's tomatoes. We can assume independence between tomatoes in the sample. What is the probability that the mean weight from the sample of 16 tomatoes xˉ is within 6 g of the population mean?

.9629

Suppose that 15% of the 1750 students at a school have experienced extreme levels of stress during the past month. A high school newspaper doesn't know this figure, but they are curious what it is, so they decide to ask a simple random sample of 160 students if they have experienced extreme levels of stress during the past month. Subsequently, they find that 10% of the sample replied "yes" to the question. Assuming the true proportion is 15%, what is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month?

.6827

The Department of Tourism reported that in May of 2017, approximately 63% of the tourists to the Philippines were from Asia. They also noted that it was the first time that the Philippines had more than 500,000 tourists in the month of May. Suppose another organization had taken a simple random sample of 600 of the tourists in that population. Assuming that the reported 63% claim is accurate, what is the approximate probability that the other organization's results were within 2 percentage points of the Department of Tourism's results?

.0548

The United Kingdom Forestry Commission reports that 43% of the 3.16 million hectares of woodland area in the United Kingdom had certification identifying them as "sustainably managed" in 2016. Suppose an employee took a simple random sample of 400 of the hectares and saw that the records showed that 47% of the sampled hectares had that certification in 2016. Assuming that the Forestry Commission's report is accurate, what is the approximate probability that more than 47% of the sample would have had the certification in 2016?

.8703

The school board claims that 85% of the over 1,600 families with children in their district send their children to public school. Skeptical of this claim, a private school administrator takes a simple random sample of 120 families with children in the district and asks them whether they send their children to public school. Assuming the school board's claim is correct, what is the probability that the administrator's results are within 5 percentage points of the school board's 85% claim?