Straighterline Introduction to Statistics| 58 questions| with complete solutions

Straighterline Introduction to Statistics| 58
questions| with complete solutions
 Course
 Statistics
Question 1: Mean Calculation
Q: The scores on a test were: 85, 92, 78, 88, 94. What is the mean score?
A:
To calculate the mean:
Mean=Sum of all scoresNumber of scores=85+92+78+88+945=4375=87.4\text{Mean} = \frac{\
text{Sum of all scores}}{\text{Number of scores}} = \frac{85 + 92 + 78 + 88 + 94}{5} = \
frac{437}{5} = 87.4Mean=Number of scoresSum of all scores=585+92+78+88+94=5437=87.4
So, the mean score is 87.4.


Question 2: Median of a Set of Numbers
Q: Find the median of the data set: 12, 7, 19, 8, 15.
A:
First, arrange the data in ascending order: 7, 8, 12, 15, 19.
The median is the middle number in the ordered set.
Thus, the median is 12.


Question 3: Mode of a Data Set
Q: Find the mode of the data set: 5, 9, 12, 5, 7, 5, 8.
A:
The mode is the number that appears most frequently.
In this set, 5 appears three times, which is more frequent than any other number.
Thus, the mode is 5.


Question 4: Standard Deviation Calculation
Q: Find the standard deviation of the data set: 10, 12, 8, 14, 16.
A:
Step 1: Find the mean:
Mean=10+12+8+14+165=605=12\text{Mean} = \frac{10 + 12 + 8 + 14 + 16}{5} = \frac{60}
{5} = 12Mean=510+12+8+14+16=560=12

,Step 2: Calculate each deviation from the mean, square it, and then average those squared
deviations:
(10−12)2=(−2)2=4(10 - 12)^2 = (-2)^2 = 4(10−12)2=(−2)2=4 (12−12)2=(0)2=0(12 - 12)^2 =
(0)^2 = 0(12−12)2=(0)2=0 (8−12)2=(−4)2=16(8 - 12)^2 = (-4)^2 = 16(8−12)2=(−4)2=16
(14−12)2=(2)2=4(14 - 12)^2 = (2)^2 = 4(14−12)2=(2)2=4 (16−12)2=(4)2=16(16 - 12)^2 = (4)^2
= 16(16−12)2=(4)2=16
Step 3: Find the variance (average of squared deviations):
Variance=4+0+16+4+165=405=8\text{Variance} = \frac{4 + 0 + 16 + 4 + 16}{5} = \frac{40}
{5} = 8Variance=54+0+16+4+16=540=8
Step 4: Take the square root of the variance to get the standard deviation:
Standard Deviation=8≈2.83\text{Standard Deviation} = \sqrt{8} \approx
2.83Standard Deviation=8≈2.83
Thus, the standard deviation is approximately 2.83.


Question 5: Probability of a Single Event
Q: What is the probability of drawing a red card from a standard deck of 52 cards?
A:
There are 26 red cards in a standard deck (13 hearts and 13 diamonds).
The probability of drawing a red card is:
P(Red)=Number of red cardsTotal number of cards=2652=12P(\text{Red}) = \frac{\
text{Number of red cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}
{2}P(Red)=Total number of cardsNumber of red cards=5226=21
Thus, the probability is 1/2 or 50%.


Question 6: Probability of Complementary Events
Q: If the probability of an event is 0.3, what is the probability of the complementary event?
A:
The probability of the complementary event is:
P(Complement)=1−P(Event)=1−0.3=0.7P(\text{Complement}) = 1 - P(\text{Event}) = 1 - 0.3 =
0.7P(Complement)=1−P(Event)=1−0.3=0.7
Thus, the probability of the complementary event is 0.7.

, Question 7: Binomial Distribution
Q: A coin is flipped 5 times. What is the probability of getting exactly 3 heads?
A:
This is a binomial probability problem. The probability of heads on each flip is 0.5. We can use
the binomial probability formula:
P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}P(X=k)=(kn)pk(1−p)n−k
Where:
 n=5n = 5n=5 (number of trials)
 k=3k = 3k=3 (number of successes, heads)
 p=0.5p = 0.5p=0.5 (probability of heads)
First, calculate (53)\binom{5}{3}(35):
(53)=5!3!(5−3)!=5×42×1=10\binom{5}{3} = \frac{5!}{3!(5 - 3)!} = \frac{5 \times 4}{2 \times
1} = 10(35)=3!(5−3)!5!=2×15×4=10
Now substitute:
P(X=3)=10×(0.5)3×(0.5)2=10×18×14=10×132=1032=0.3125P(X = 3) = 10 \times (0.5)^3 \times
(0.5)^2 = 10 \times \frac{1}{8} \times \frac{1}{4} = 10 \times \frac{1}{32} = \frac{10}{32} =
0.3125P(X=3)=10×(0.5)3×(0.5)2=10×81×41=10×321=3210=0.3125
Thus, the probability of getting exactly 3 heads is 0.3125.


Question 8: Z-Score Calculation
Q: What is the Z-score of a data point of 85, if the mean is 80 and the standard deviation is 5?
A:
The formula for the Z-score is:
Z=X−μσZ = \frac{X - \mu}{\sigma}Z=σX−μ
Where:
 X=85X = 85X=85 (data point)
 μ=80\mu = 80μ=80 (mean)
 σ=5\sigma = 5σ=5 (standard deviation)
Z=85−805=55=1Z = \frac{85 - 80}{5} = \frac{5}{5} = 1Z=585−80=55=1
Thus, the Z-score is 1.