Unit 6 Test: Lessons 45–56 Solutions

#	Answer	Lesson Origin
1)	C	45
2)	D	48
3)	B	55
4)	D	52
5)	C	45
6)	D	53
7)	A	49
8)	B	56
9)	A	54
10)	B	46
11)	B	48
12)	C	51
13)	B	56
14)	D	52
15)	C	50
16)	A	46
17)	C	50
18)	A	54
19)	A	47
20)	D	51
21)	B	53
22)	D	49
23)	C	47
24)	A	46
25)	D	55

Determine the type of distribution for the data set:
$\{6, 8, 8, 9, 9, 15, 29\}$

Left-skewed
Symmetric
Right-skewed
Cannot be determined

$Mean : \frac{6 + 8 + 8 + 9 + 9 + 15 + 29}{7} = 12 M edian : 9 M ean > M edian R ight ‐ skewed$

Note

The mean is less than the median for left-skewed distributions.
The mean and median are equal for symmetric distributions.

Callie’s Coffee is a drive-thru store. They decide to survey every tenth customer about their ordering experience. Identify the type of sampling method used.

Cluster sampling
Simple random sampling
Stratified sampling
Systematic sampling

Systematic: After the population is randomized to avoid bias, a starting point is determined from the population, and then every $n^{th}$ individual is selected.

Note

The scenario does not reflect the other types of sampling.

A coin is flipped four times. What is the probability of the coin landing heads up exactly two times?

$\frac{1}{2}$
$\frac{3}{8}$
$\frac{1}{4}$
$\frac{1}{16}$

$Total : 2^{4} = 16 (H, H, T, T), (H, T, H, T), (H, T, T, H), (T, H, H, T), (T, H, T, H), (T, T, H, H) P (exactly two heads) = \frac{6}{16} = \frac{3}{8}$

Note

This option is the probability of one coin landing heads up.
This option is the probability of two coins landing heads up.
This option is the probability of any combination of four coins being flipped.

What expression represents the sum of the 49th row of Pascal’s Triangle?

$49^{49}$
49!
$n^{49}$
$2^{49}$

The sum of n-rows of Pascal’s Triangle is $2^{n} .$

$n = 49$

Note

This option is $n^{n}$ .
This option is n!.
This option does not result in a value.

Heights of privacy hedges are normally distributed and grown to an average height of 65 inches with a standard deviation of 3 inches. What is the range that includes 95% of the heights of the privacy hedges?

62 to 68 inches
56 to 74 inches
59 to 71 inches
62 to 71 inches

$μ - 2 σ < μ < μ + 2 σ 65 - 2 (3) < μ < 65 + 2 (3) 59 < μ < 71$

Note

This option is one standard deviation.
This option is three standard deviations.
This option is one standard deviation below and two standard deviations above the mean.

A bag contains four red, three yellow, and five green marbles. What is the probability of selecting a marble that is not yellow?

$\frac{1}{3}$
$\frac{1}{4}$
$\frac{5}{12}$
$\frac{3}{4}$

$Total : 4 + 3 + 5 = 12 P (Y) = \frac{3}{12} = \frac{1}{4} 1 - \frac{1}{4} = \frac{3}{4}$

Note

This option is the probability of selecting a red marble.
This option is the probability of selecting a yellow marble.
This option is the probability of selecting a green marble.

A survey of 225 community members has a sample standard deviation of 5.1 percentage points. Determine the maximum error of the estimate for a 90% confidence level.

0.56
0.67
0.88
1.65

$E = (1.645) (\frac{5.1}{\sqrt{225}}) = 0.5593$

Note

This option is the value for a 95% confidence level.
This option is the value for a 99% confidence level.
This option is the rounded value of the z-score.

What does $P (B | A)$ mean when working with conditional probabilities?

The probability of A given B occurred.
The probability of B given A occurred.
The probability of A and B occurred.
The probability of A or B occurred.

Note

This option is $P (A | B)$ .
This option is $P (A \cap B)$ .
This option is $P (A \cup B)$ .

Which diagram represents $A \cup B ?$

A or B means all the information is included in either set A or set B.

Note

This option is $A \cap B$ .
This option is $A \cap B ’$ .
This option is $(A \cup B) ’$ .

Mr. Boone noticed the final exam for his class was normally distributed with an average of 82% and a standard deviation of 5 percentage points. What is the approximate likelihood that a student scored higher than 91%?

1.8%
3.6%
87%
96.4%

$z = \frac{91 - 82}{5} = 1.8 z ‐ table : 0.9641 1 - 0.9641 = 0.0359 0.0359 (100) = 3.59 %$

Note

This option is the z-score instead of the z-table value.
This option is one standard deviation from the mean.
This option is the likelihood that a student scores lower than 91%.

What is the main problem with a clothing store only surveying the first ten customers that entered the store on a single day?

The population is unknown.
The sample is too small.
The sample is too large.
The sample is not random.

Small Sample: The sample is too small to draw any valid conclusions about the population $(n > 30) .$

Judges ranked seven auditioning violinists for the Town Symphony’s first- and second-chair violin seats. How many possible ways could the judges rank the violinists first and second choice?

14
21
42
49

$n P r (7, 2) = \frac{7!}{(7 - 2)!} = \frac{7 \cdot 6 \cdot 5!}{5!} = 42$

Note

This option is the product of seven and two.
This option is the answer if the problem was a combination.
This option is seven-squared.

Fitness-4-All Gym documented that 70% of members use the cardio machines and 50% lift weights. If $P (cardio | weights) = 0.8,$ what is $P (cardio \cap weights) ?$

0.80
0.56
0.40
0.35

Note

This option is the conditional probability.
This option is the product of 0.8 and 0.5.
This option is the product of 0.7 and 0.5.

$\begin{array}{rcl} P (cardio \cap weights) & = & P (cardio) \cdot P (cardio | weights) \\ = & (0.7) (0.8) = 0.56 \end{array}$

Determine the coefficient of the middle term when the binomial ${(3 x + 1)}^{4}$ is expanded.

4
6
9
54

$Row 4 : 1, 4, 6, 4, 1 6 {(3 x)}^{2} {(1)}^{2} = 6 (9 x^{2}) (1) = 54 x^{2}$

Note

This option is the exponent and row.
This option is the coefficient ignoring the term 3x.
This option is ${(3 x)}^{2}$ .

Athletes were divided into two groups to determine if breathing exercises improved performance on game days. After a simulation of 10,000 trials, the standard deviation was 1.5 percentage points.
Determine the z-score for the observed difference at the 95% level.

1.50
1.96
2.67
3.40

$Observed difference z = \frac{72.2 - 68.6}{1.5} = 2.266$

Group n = 75	Average Performance
Treatment: added breathing exercises	72.2%
Control: no breathing exercises	68.8%

Note

This option is the standard deviation.
This option is the z-score for a 95% confidence level.
This option is the observed difference of the means.

A normal data distribution has a mean of 100 and a standard deviation of 15. What approximate percentage of data falls below 87?

19%
34%
68%
81%

$z = \frac{87 - 100}{15} = - 0.87 z ‐ table : 0.1922$

Note

This option is the percentage between the mean and one standard deviation above the mean.
This option is the percentage within one standard deviation of the mean.
This option is the percentage if the negative symbol of the z-score is ignored.

A researcher calculated the z-score of the observed difference between treatment and control groups to be 0.34. What can the researcher conclude?

The treatment is highly effective.
A larger experiment should be conducted.
The results are not statistically significant.
The results are statistically significant.

The closer the z-score is to zero, the less statistically significant it is.

Note

A, D) A highly effective treatment would have a z-score > 1.96.

This option cannot be determined from the information.

From a standard deck of cards, determine P(king or heart).

$\frac{4}{13}$
$\frac{17}{52}$
$\frac{9}{26}$
$\frac{1}{4}$

Note

This option is the probability if the king of hearts is double-counted.
This option is the probability if all the fractions are added together.
This option is the probability of selecting a card with hearts.

$\begin{array}{rcl} P (king or heart) & = & 4 kings + 13 hearts - king of hearts \\ = & \frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13} \end{array}$

A scientist collects 200 crayfish from Lake Luna containing 10,000 crayfish. Name the sample.

The 200 crayfish collected.
All crayfish in Lake Luna.
All crayfish in all lakes.
There is no sample named.

Sample: A subset (or part) of a population from which data is collected.

Note

This option is the population of crayfish in Lake Luna.
This option is the world population of crayfish.

A restaurant designed a new menu with eight appetizers, six salads, and twelve entrees. How many unique meals can be created when one option from each part of the menu is selected?

26
72
96
576

$8 \cdot 6 \cdot 12 = 576$

Note

This option is the sum of the options.
This option is the total number of options without selecting an appetizer.
This option is the total number of options without selecting a salad.

If an experiment runs 200 times with a theoretical probability of 0.25, how many successes can you expect?

25
50
100
200

$(200) (0.25) = 50$

Note

This option would be the number of successes for 100 trials.
This option would be the number of successes if the probability is 0.5.
This option is the number of trials in the experiment.

A consumer group randomly selected 150 customers to complete a survey. If 9% of customers reported being dissatisfied with a margin of error equal to ±7%, what interval contains the population parameter for satisfied customers?

Between 2% and 16% of customers will be satisfied.
Between 91% and 98% of customers will be satisfied.
Between 84% and 91% of customers will be satisfied.
Between 84% and 98% of customers will be satisfied.

$100 - 9 = 91 91 - 7 = 84 91 + 7 = 98$

Note

This option is the range of dissatisfied customers.

B, C) This option is only half of the margin of error.

A researcher surveys and records the species of birds people have seen when visiting a park. What type of study is this?

Experiment
Observation
Survey
Census

Sample Survey: Through researcher questionnaires or interviews, conclusions are drawn about the population using the sample.

Note

The scenario does not reflect the other types of studies.

In a standard normal distribution, approximately what proportion of data falls between $z = - 0.1$ and $z = 1 .$

0.38
0.46
0.68
0.84

$z = - 0.1 z ‐ table : 0.4602 z = 1 z ‐ table : 0.8413 0.8413 - 0.4602 = 0.3811$

Note

This option is the z-score for $- 0.1 .$
This option is the proportion within one standard deviation.
This option is the z-score for 1.

A six-sided number cube is rolled twice. What is the probability that the first number rolled is even, and the next number is less than three?

$\frac{5}{6}$
$\frac{1}{2}$
$\frac{1}{3}$
$\frac{1}{6}$

$Even : \{2, 4, 6\} Less than 3 : \{1, 2\} P (even, > 3) = \frac{3}{6} \cdot \frac{2}{6} = \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6}$

Note

This option is the probability if you add the fractions.
This option is the probability of rolling an even number.
This option is the probability of rolling a number less than three.