Question 1

Assume a data set is normally distributed with mean 129 and standard deviation 35. If the data set
Contains 350 data values, approximately how many of the data values will fall within the range 59 to
199?

Accepted Answer

To solve the problem, first we standardize the range 59 to 199 using the formula:
z = (x - μ) / σ

For the lower bound, we have:
z = (59 - 129) / 35 = -2

For the upper bound, we have:
z = (199 - 129) / 35 = 2

We can then use a standard normal distribution table or calculator to find the area under the curve between -2 and 2, which is approximately 0.9545.

Finally, we multiply this area by the total number of data values to get:
0.9545 x 350 = 334.18

Since we can't have a fraction of a data value, we round to the nearest whole number and get 334. Therefore, the answer is C. We can use the empirical rule to estimate the proportion of data values that fall within the range of 59 to 199. The empirical rule states that approximately 68% of data values fall within one standard deviation of the mean, 95% within two standard deviations of the mean, and 99.7% within three standard deviations of the mean for normally distributed data.

To apply this rule, we first need to find the z-scores for the values 59 and 199:

z1 = (59 - 129) / 35 = -2
z2 = (199 - 129) / 35 = 2

Next, we find the area under the standard normal distribution curve between these two z-scores:

area = P(-2 ≤ Z ≤ 2) ≈ 0.9545

Finally, we can estimate the proportion of data values in the range of 59 to 199 by multiplying this area by the total number of data values:

proportion = 0.9545 * 350 ≈ 334

Therefore, approximately 334 data values will fall within the range of 59 to 199. The best choice is C.

Question 2

If a data value has a Z score of , then half of the data are below that value, and half are above.

Accepted Answer

The answer of If a data value has a Z...

Question 3

Assume a data set is normally distributed with mean 17 and standard deviation 1.5. If the data set
Contains 40 data values, approximately how many of the data values will fall within the range 15.5
To 18.5?

Accepted Answer

The answer of Assume a data set is normally distributed...

Question 4

If a student scored 1,484 on the SAT test which has a mean of 1,026 and a standard deviation of 210
and then scored 126 on an IQ test with a mean of 100 and a standard deviation of 15, determine
which test she did better on and explain your reasoning.

Accepted Answer

The answer of If a student scored 1,484 on the...

Question 5

When data are normally distributed, approximately of the values are within 1 standard
deviation of the mean, approximately are within 2 standard deviations of the mean, and
approximately are within 3 standard deviations of the mean.

Accepted Answer

The answer of When data are normally distributed, approximately of...

Question 6

Suppose that you collect a very large number of maple tree seed pods, record their lengths and draw
A histogram of the data using narrow class widths. Next you collect a very large number of adult
Women's heights and again draw a histogram using narrow classes. How would you expect the two
Histograms to compare?

Accepted Answer

Both sets of data are likely to follow a normal distribution. While the units of measurement are different between the two data sets, this should not affect the shape of the histograms. Additionally, using narrow classes helps to better approximate a normal distribution. Both sets of data are continuous and could approximate a normal distribution. Narrow class widths would help smooth out any irregularities in the data and make the histograms more bell-shaped. It is also unlikely that the mean value of the heights would be significantly larger than the mean value of the maple seed pod lengths.

Question 7

The data below are the number of cattle on farms in the United States (in millions)for each year that
Begins a decade from 1910 to 2010. Draw a box plot for the data below. Are there any outliers? If
So, what are they?

Accepted Answer

The answer of The data below are the number of...

Question 8

Assume that the finishing times for a group of runners in a marathon race are likely to be normally
distributed. The mean finish time for male marathon runners is 288.46 min with a standard deviation
of 39.7 min. The mean finish time for female marathon runners is 305.59 min with a standard
deviation of 39.6 min. Would a male with a time of 128.62 minutes or a female with a time of
138.95 minutes finish faster relative to their gender? Explain why. (Remember that in a race you
want to finish in a short time.)

Accepted Answer

The answer of Assume that the finishing times for a...

Question 9

Normal distributions are very commonly used for quality control in manufacturing. For example, a
Manufacturer of light bulbs would surely be interested in studying how long their bulbs last. In one
Case, the lifetime of one model of light bulb is normally distributed and has a mean X = 1,900 hours
And a standard deviation of s = 250 hours. What is the probability that a bulb will last between 1,650
And 1,900 hours?

Accepted Answer

Since the lifetime of the bulb is normally distributed with a mean of 1,900 hours and a standard deviation of 250 hours, we can standardize the values for 1,650 hours and 1,900 hours using the formula z = (x - mean)/standard deviation.
z for 1,650 hours = (1,650 - 1,900)/250 = -1
z for 1,900 hours = (1,900 - 1,900)/250 = 0
Using a standard normal distribution table or calculator, we can find the area under the curve between z = -1 and z = 0, which is approximately 0.341. Therefore, the probability that a bulb will last between 1,650 and 1,900 hours is 0.341 or 34.1%. The correct answer is B. We need to find the probability that the bulb will last between 1,650 and 1,900 hours. We can use the standard normal distribution by standardizing the values first:

z1 = (1650-1900)/250 = -1
z2 = (1900-1900)/250 = 0

Using a standard normal table or a calculator, we can find the probability that a standard normal distribution is between -1 and 0 is 0.341. Therefore, the probability that the bulb will last between 1,650 and 1,900 hours is 0.341 or 34.1%. The correct answer is B.

Question 10

Find the area under the normal distribution curve between z = 0 and z = -1.35.

Accepted Answer

The answer of Find the area under the normal distribution...

Question 11

Assume that males have a mean height of 69.1 in. with a standard deviation of 2.9 in., and females
Have a mean height of 63.7 in. with a standard deviation of 2.7 in. Based on Z scores, would a 5'6"
Male or a 5'2" female be taller compared to the rest of their gender?

Accepted Answer

To compare the two heights based on their respective genders, we need to convert them to Z-scores.

For the 5'6" male: 
1. Convert 5'6" to inches: 5 ft * 12 in/ft + 6 in = 66 in 
2. Calculate the Z-score: 
(Z-score) = (observed value - population mean) / standard deviation 
= (66 - 69.1) / 2.9 
= -1.07

For the 5'2" female: 
1. Convert 5'2" to inches: 5 ft * 12 in/ft + 2 in = 62 in 
2. Calculate the Z-score: 
(Z-score) = (observed value - population mean) / standard deviation 
= (62 - 63.7) / 2.7 
= -0.63

A negative Z-score means the observed height is below the average for their gender. Therefore, the female with a Z-score of -0.63 (5'2" height) is less below the average height for females than the male with a Z-score of -1.07 (5'6" height) compared to the average height for males. Therefore, the female is relatively taller compared to the rest of their gender.

Question 12

Suppose the mean number of candies in a standard vending machine bag is 47.6, with a standard
deviation of 1.3. Assume that the number of candies per package is normally distributed.
a. Fill in the blanks below the normal curve with the appropriate information.  b. If you buy 1,000 packages of candies, about how many bags would you expect to have between
46 and 49 M&M's in it?
c. What percentage of the bags have less than 45 candies?

Accepted Answer

The answer of Suppose the mean number of candies in...

Question 13

A Z score tells you how many a data value is from the mean. Data values that are below the
mean would have Z scores.

Accepted Answer

The answer of A Z score tells you how many...

Question 14

Find the area under the normal distribution curve between z = 0 and z = 2.35.

Accepted Answer

The answer of Find the area under the normal distribution...

Question 15

A standardized test of intelligence is scaled so that the mean IQ is 100, and the standard deviation is
15. On a standardized IQ test, what is the Z score for a person with an IQ of 145?

Accepted Answer

To find the Z score, we use the formula Z = (X - µ) / σ, where X is the IQ score, µ is the mean IQ, and σ is the standard deviation. Plugging in the given values, we get Z = (145 - 100) / 15 = 3. Therefore, the Z score for a person with an IQ of 145 is 3. Choice C is the only option that matches this result. The formula for calculating Z score is (X - μ) / σ, where X is the individual's score, μ is the mean, and σ is the standard deviation. Plugging in the values given in the question, we get (145 - 100) / 15 = 3. Therefore, the Z score for a person with an IQ of 145 is 3.

Question 16

Given the following information about test scores, draw a box plot for the data.

Accepted Answer

The answer of Given the following information about test scores,...

Question 17

Find the area under the normal distribution curve between z = 0.65 and z = 2.30.

Accepted Answer

To find the area under the normal distribution curve between two z-scores, you subtract the area to the left of the lower z-score from the area to the left of the higher z-score. Looking up z = 0.65 in the standard normal distribution table gives an area of 0.7422 to the left, and z = 2.30 gives an area of 0.9893 to the left. Subtracting these values gives 0.9893 - 0.7422 = 0.247.

Question 18

Normal distributions are very commonly used for quality control in manufacturing. For example, a
Manufacturer of light bulbs would surely be interested in studying how long their bulbs last. In one
Case, the lifetime of one model of light bulb is normally distributed and has a mean X = 1,800 hours
And a standard deviation of s = 250 hours. What percentage of bulbs would you expect to last longer
Than 2,550 hours?

Accepted Answer

We need to find the percentage of bulbs that last longer than 2,550 hours.

We can standardize the variable using the formula:

z = (x - μ) / σ

where: 
x = 2,550 hours 
μ = 1,800 hours 
σ = 250 hours

z = (2,550 - 1,800) / 250 = 3

Using a standard normal distribution table or calculator, we can find that the percentage of values that fall above z = 3 is approximately 0.1%. Therefore, we can expect that only 0.1% of bulbs will last longer than 2,550 hours.

Question 19

Suppose that the mean sale price for existing homes in the United States in is $219,300. Assume
that sale prices are normally distributed with a standard deviation of $44,000.
a. Fill in the blanks below the normal curve with the appropriate information.  Find the percentage of existing homes that:
b. sold for between $131,300 and $307,300
c. sold for more than $263,300

Accepted Answer

The answer of Suppose that the mean sale price for...

Question 20

According to recent research, the mean height for women in the U.S. is 63.7 inches (which you
Already knew, right?), and the standard deviation is 2.7 inches. The normal graph below is marked
With the mean and heights that are one and two standard deviations above and below the mean.  Make an educated guess on the percentage of total shaded area between 58.3 and 69.1 inches.

Accepted Answer

Using the empirical rule, we know that approximately 68% of the data falls within one standard deviation of the mean and approximately 95% falls within two standard deviations of the mean.

So, the interval from 58.3 to 69.1 inches is within two standard deviations of the mean. Therefore, we can estimate that approximately 95% of the total shaded area is between these two heights. According to the Empirical Rule, approximately 68% of the data falls within one standard deviation of the mean, 95% of the data falls within two standard deviations of the mean, and 99.7% of the data falls within three standard deviations of the mean. Since we are looking for the percentage of area between 58.3 and 69.1 inches, which is within two standard deviations of the mean, the best choice is B, about 95%.

Quiz 11: Statistics

Assume a data set is normally distributed with mean 129 and standard deviation 35. If the data set Contains 350 data values, approximately how many of the data values will fall within the range 59 to 199?

If a data value has a Z score of , then half of the data are below that value, and half are above.

Assume a data set is normally distributed with mean 17 and standard deviation 1.5. If the data set Contains 40 data values, approximately how many of the data values will fall within the range 15.5 To 18.5?

If a student scored 1,484 on the SAT test which has a mean of 1,026 and a standard deviation of 210 and then scored 126 on an IQ test with a mean of 100 and a standard deviation of 15, determine which test she did better on and explain your reasoning.

When data are normally distributed, approximately of the values are within 1 standard deviation of the mean, approximately are within 2 standard deviations of the mean, and approximately are within 3 standard deviations of the mean.

The data below are the number of cattle on farms in the United States (in millions) for each year that Begins a decade from 1910 to 2010. Draw a box plot for the data below. Are there any outliers? If So, what are they?

Find the area under the normal distribution curve between z = 0 and z = -1.35.

Assume that males have a mean height of 69.1 in. with a standard deviation of 2.9 in., and females Have a mean height of 63.7 in. with a standard deviation of 2.7 in. Based on Z scores, would a 5'6" Male or a 5'2" female be taller compared to the rest of their gender?

A Z score tells you how many a data value is from the mean. Data values that are below the mean would have Z scores.

Find the area under the normal distribution curve between z = 0 and z = 2.35.

A standardized test of intelligence is scaled so that the mean IQ is 100, and the standard deviation is 15. On a standardized IQ test, what is the Z score for a person with an IQ of 145?

Given the following information about test scores, draw a box plot for the data.

Find the area under the normal distribution curve between z = 0.65 and z = 2.30.