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 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 shipment of oranges has a population mean weight of $150$ grams and a standard deviation of $6$ grams. For a random sample of $n=64$ oranges, what is the probability that the sample mean weight is less than $150$ grams or greater than $151.5$ grams? Would a sample mean above $151.5$ grams be considered unusual?

A weather balloon records the following pairs of altitude in thousands of feet and temperature in ${}^{\circ }F$: $(1.5,60)$, $(3.8,52)$, $(6.0,44)$, $(8.2,35)$, and $(10.5,28)$. Find (a) the explained variation, (b) the unexplained variation.

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 between $-5\%$ and $10\%$?

The following are the cholesterol levels (in $\text{mg/dL}$) of $10$ randomly selected adults from a health survey. Assume the cholesterol levels are normally distributed. Construct a $99\%$ confidence interval for the population standard deviation.

Data: $180,190,195,200,210,220,225,230,240,250$

A health study estimates the standard deviation of daily step counts as $\sigma =9.8$. How many participants are required to estimate the mean daily steps within $2$ steps with $99\%$ confidence?

True or false: If samples of size $n=5$ are drawn from a highly skewed population with finite variance, the distribution of the sample mean, $\bar{X}$, is approximately normal.

A researcher wants to compare the proportion of urban and rural residents who recycle regularly. She wants her estimate of the difference to be within $2$ percentage points with $95\%$ confidence. Prior studies suggest that $35\%$ of urban and $27\%$ of rural residents recycle regularly. What is the minimum sample size she should use for each group?

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?

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.

If a random sample of size $n$ is drawn from a population that is exactly normal with mean $\mu$ and variance ${\sigma }^2$, which statement is correct about the sampling distribution of the sample mean $\bar{X}$?