Sampling Distribution of the Sample Mean and Central Limit Theorem: Videos & Practice Problems

A botanist is studying the height of a rare plant species. Out of a population of $150$ plants, the mean height is $32.4\text{ cm}$ with a standard deviation of $8.1\text{ cm}$. If a random sample of $40$ plants is chosen, what is the probability that the sample mean height is between $30\text{ cm}$ and $35\text{ cm}$? Should the finite correction factor be used?

The mean daily temperature in Helsinki during winter months has a mean of ${2.8}^{\circ }\text{C}$ and a standard deviation of ${5.7}^{\circ }\text{C}$. Suppose random samples of size $64$ are drawn from this population, and the mean temperature of each sample is recorded. What are the mean and standard deviation of the sampling distribution of sample means?

The weights of packages shipped by a logistics company are normally distributed with a mean of $12.0\text{ kg}$ and a standard deviation of $2.4\text{ kg}$. Random samples of $42$ packages are taken and the mean of each sample is recorded. What are the mean and standard deviation of the sampling distribution of sample means?

A researcher measures the resting heart rates of a random sample of $36$ athletes and finds a sample mean of $62.5$ beats per minute with a sample standard deviation of $7.9$ beats per minute. Construct a $95\%$ confidence interval for the population mean resting heart rate.

A factory packages bags of flour with an average weight of $2.02$ kg and a standard deviation of $0.03$ kg. The acceptable weight range for each bag is $1.95$ kg to $2.05$ kg. You randomly select $10$ bags of flour. What is the probability that you select at least one bag with a weight within the acceptable range?

A company produces small bottles of essential oil with an average fill amount of $12.02$ ml and a standard deviation of $0.04$ ml. You randomly select a sample of $5$ bottles and measure the mean fill amount. What is the probability that the sample mean is within the acceptable range of $11.95$ ml to $12.05$ ml?

A company’s filling machine dispenses essential oil with a mean of $11.96$ ml and a standard deviation of $0.05$ ml. A quality inspector randomly selects three separate samples, each containing five bottles, and measures the sample mean fill amount for each. What is the probability that at least one of these three samples has a sample mean within the acceptable range of $11.90$ ml to $12.00$ ml?

A packaging machine fills small boxes with herbal powder. The filling weight has a mean of $24.8$ grams and a standard deviation of $0.10$ grams. A quality control inspector is comparing two types of inspections:

Type A: Measuring the fill weight of a single box.

Type B: Measuring the mean fill weight of a sample of $4$ boxes.

Which type of inspection is more sensitive to a shift in the filling process—an individual measurement or the sample mean?

The average annual water consumption in a city since $2000$ is normally distributed with a mean of $4200$ million gallons and a standard deviation of $210$ million gallons. If a random sample of $4$ years is chosen, what is the probability that the sample mean water consumption is less than $4000$ million gallons?

The average annual water consumption in a city since $2000$ is normally distributed with a mean of $4200$ million gallons and a standard deviation of $210$ million gallons. If a random sample of $4$ years is chosen, what is the probability that the sample mean water consumption is between $4350$ and $4500$ million gallons?

The average annual water consumption in a city since $2000$ is normally distributed with a mean of $4200$ million gallons and a standard deviation of $210$ million gallons. If a random sample of $4$ years is chosen, what is the probability that the sample mean water consumption is greater than $4500$ million gallons?

The mean daily rainfall in a rainforest region is normally distributed with a mean of $18.2\text{ mm}$ and a standard deviation of $6.3\text{ mm}$. If a sample of $10$ days is randomly chosen, what is the probability that the sample mean rainfall is between $14.4$ and $21.7\text{ mm}$?

The mean daily rainfall in a rainforest region is normally distributed with a mean of $18.2\text{ mm}$ and a standard deviation of $6.3\text{ mm}$. If a sample of $10$ days is randomly chosen, what is the probability that the sample mean rainfall is greater than $22.8\text{ mm}$?

The mean SAT math score in a certain year is $530$. A random sample of $42$ SAT math scores is taken. What is the probability that the sample mean is less than $500$? Assume $\sigma =75$.

The mean SAT math score in a certain year is $530$. A random sample of $42$ SAT math scores is taken. What is the probability that the sample mean is between $527$ and $532$? Assume $\sigma =75$.

A machine fills bottles with a mean volume of $500\text{ mL}$ and a standard deviation of $5\text{ mL}$. If a random sample of $36$ bottles is selected, what is the probability that the sample mean volume is less than $498.5\text{ mL}$?

A machine fills bottles with a mean volume of $500\text{ mL}$ and a standard deviation of $5\text{ mL}$. If a random sample of $36$ bottles is selected, what is the probability that the sample mean volume is more than $501.7\text{ mL}$?

A company reports that the mean monthly rent for a one-bedroom apartment in a city is $\$1350$ with a standard deviation of $\$125$. If a random sample of $48$ apartments is selected, what is the probability that the sample mean rent is less than $\$1390$?

A company reports that the mean monthly rent for a one-bedroom apartment in a city is $\$1350$ with a standard deviation of $\$125$. If a random sample of $48$ apartments is selected, what is the probability that the sample mean rent is more than $\$1320$?

A company claims that the average delivery time for its packages is $48$ hours with a standard deviation of $4$ hours. If a random sample of $32$ deliveries is selected, what is the probability that the sample mean delivery time is less than $46$ hours?

A certain stock's daily percent return on Fridays has a mean of $3.12\%$ and a standard deviation of $41.25\%$. If random samples of $40$ days are selected and the mean return for each sample is calculated, what is the probability that a sample mean is greater than $17\%$?

The time (in minutes) it takes to assemble a product is left-skewed with a mean of $20$ and a standard deviation of $6$. If random samples of size $36$ are selected, what is the shape of the sampling distribution of the sample mean?

A university reports that the average annual tuition increase over the past $40$ years has been $\$1,200$, with a standard deviation of $\$2,400$. If a random sample of $36$ years is selected, what is the probability that the sample mean annual tuition increase is between $\$900$ and $\$1,500$?

A factory produces light bulbs with a mean lifetime of $1200$ hours and a standard deviation of $100$ hours. If a quality control manager selects a random sample of $25$ bulbs, what is the probability that the sample mean lifetime is greater than $1230$ hours?

The weights of apples in an orchard are normally distributed. Is it more probable to randomly pick an apple weighing less than $120$ grams, or to randomly select a group of $8$ apples whose average weight is less than $120$ grams?