Question 1

A six-year study in a certain country found that birth weights of newborns were normally distributed, with a mean of 3758 grams and a standard deviation of 496 grams. What is the z-score for a newborn weighing 4000 grams?

Accepted Answer

The answer of A six-year study in a certain country...

Question 2

To find the confidence interval, adjust the result of the poll by adding and subtracting the ______________.

Accepted Answer

The margin of error represents the amount by which the poll result could be off due to random sampling variation. Adding and subtracting the margin of error to the poll result creates a range of values within which the true population parameter (e.g. proportion or mean) is likely to lie with a certain level of confidence. Thus, the confidence interval can be found by using the margin of error.

Question 3

For a 95% level of confidence and a sample size of n, the margin of error is approximately:

Accepted Answer

The answer of For a 95% level of confidence and...

Question 4

It is found that the number of raisins in a box of a popular cereal is normally distributed, with a mean of 142 raisins per box and a standard deviation of 11 raisins. Your cereal box has 159 raisins. What is the z-score for this box of cereal?

Accepted Answer

The answer of It is found that the number of...

Question 5

It is known that in the absence of treatment, 68% of the patients with a certain illness will improve. The central limit theorem tells us that the percentages of patients in groups of 500 that improve in the absence of treatment are approximately normally distributed. Find the mean of the normal distribution given by the central limit theorem.

Accepted Answer

The mean of the normal distribution given by the central limit theorem is equal to the mean of the original distribution, which is 68%. Therefore, the answer is D) 68%.

Question 6

The _______________ for a number relative to a list of data is the percentage of data points that are less than or equal to that number.

Accepted Answer

The definition given matches the definition of a percentile. A z-score is a measure of how many standard deviations a value is from the mean, a quartile divides a data set into four equal parts, and a t-score is similar to a z-score but is used when the sample size is small.

Question 7

For data points above the mean, the z-score is _______________.

Accepted Answer

The z-score is a measure of how many standard deviations a data point is from the mean. If a data point is above the mean, it has a positive deviation, and therefore a positive z-score.

Question 8

Ridge counts on fingerprints are approximately normally distributed, with a mean of about 140 and a standard deviation of 50. Which of the following intervals will contain 47.5% of the population?

Accepted Answer

The answer of Ridge counts on fingerprints are approximately normally...

Question 9

Suppose we toss 150 fair coins and record the percentage of heads. According to the central limit theorem, the percentages of heads resulting from such experiments are approximately normally distributed. Find the standard deviation of this normal distribution.

Accepted Answer

The answer of Suppose we toss 150 fair coins and...

Question 10

If we construct a confidence interval from the result of a poll, then the interval should contain the true result.

Accepted Answer

A confidence interval provides a range of values within which we expect the true parameter to lie, with a certain level of confidence (e.g., 95%). However, it is not guaranteed to contain the true result; there is always a chance (equal to 100% minus the confidence level) that the true parameter lies outside this interval.

Question 11

The IQ (Intelligence Quotient) test scores are normally distributed and are scaled to a mean of 100 and a standard deviation of 15. If a person's IQ has a z-score of 1.8, what is the IQ score?

Accepted Answer

The answer of The IQ (Intelligence Quotient) test scores are...

Question 12

Ridge counts on fingerprints are approximately normally distributed, with a mean of about 140 and a standard deviation of 50. It then follows that 68% of the population will have:

Accepted Answer

The 68% figure refers to the percentage of data within one standard deviation of the mean in a normal distribution, which in this case is 140 ± 50 ridges.

Question 13

The _______________ of a poll tells the percentage of such polls in which the confidence interval includes the true result.

Accepted Answer

The confidence level of a poll is the percentage of times the true result will be captured in the margin of error. It is typically set at 95% or 99%.

Question 14

The average yearly high temperature in a certain city is recorded. It is found that the mean temperature is 70.2°F with a standard deviation of 8.2°F. Assuming that the data are normally distributed, in what range should 95% of the data lie?

Accepted Answer

For a normal distribution, 95% of the data lies within approximately 2 standard deviations of the mean. Calculating 2 standard deviations from the mean (70.2 ± 2*8.2) gives the range 53.8°F to 86.6°F.

Question 15

It is known that in the absence of treatment, 68% of the patients with a certain illness will improve. The central limit theorem tells us that the percentages of patients in groups of 500 that improve in the absence of treatment are approximately normally distributed. Find the standard deviation of the normal distribution given by the central limit theorem.

Accepted Answer

The standard deviation of the sampling distribution of a proportion is given by the formula $\sqrt{\frac{p(1-p)}{n}}$, where $p$ is the population proportion and $n$ is the sample size. Here, $p = 0.68$ and $n = 500$, so the standard deviation is $\sqrt{\frac{0.68(1-0.68)}{500}} = \sqrt{\frac{0.68 \times 0.32}{500}} = \sqrt{\frac{0.2176}{500}} = \sqrt{0.0004352} = 0.020856$, which is approximately 2.1% when converted to a percentage.

Question 16

If a set of data is normally distributed, then about _______________ of the data lie within one standard deviation below the mean.

Accepted Answer

In a normal distribution, about 34% of the data lies within one standard deviation below the mean, and another 34% lies within one standard deviation above the mean, making up approximately 68% of the data within one standard deviation of the mean.

Question 17

The average yearly high temperature in a certain city is recorded. It is found that the mean temperature is 70.2°F with a standard deviation of 8.2°F. Assuming that the data are normally distributed, in what range should 68% of the data lie?

Accepted Answer

68% of the data will lie within one standard deviation of the mean in either direction.

One standard deviation in this case is equal to 8.2°F.

Therefore, we can find the range by adding and subtracting one standard deviation from the mean:

70.2°F +/- 8.2°F = 62.0°F - 78.4°F

So, the correct answer is B.

Question 18

If a set of data is normally distributed, then about _______________ of the data lie within two standard deviations above the mean.

Accepted Answer

The answer of If a set of data is normally...

Question 19

Assume we know that 20% of Americans suffer from a certain type of allergy. Suppose we take a random sample of 8000 Americans and record the percentage who suffer from this allergy. What guarantees that percentages from such a survey will be normally distributed?

Accepted Answer

The central limit theorem guarantees that the sample means will be normally distributed, even if the population distribution is not normal, as long as the sample size is sufficiently large (usually n > 30). In this case, we have a sample size of 8000, which is more than enough to satisfy this condition. Therefore, we can assume that the percentages from this survey will be normally distributed.

Question 20

The z-score can be used to compare the relative location of values from the sets of data with different distribution. Suppose that a Statistics class gave two midterms so far. If you scored 83 on Midterm 1 and 85 on Midterm 2, which score is relatively higher than the other? Use the information below.

Accepted Answer

The answer of The z-score can be used to compare...

Quiz 6: Statistics