The Predictable Behavior of Random Events Homework
Name _____________________
1. When rolling a single 6-sided die, what is the probability of rolling a 5?
____ _______
2. When drawing from a standard deck of playing cards (which has 52 cards):
2a. What is the probability of drawing an Ace? That is, what is P(ace)?
____ ________
2b. What is P(spade)?
_____ _______
3. If our null hypothesis about the probability of a success is H0: P=0.55 then
3a. What is P(1 randomly selected value is a success)?
_____0.55_______
3b. What is P(the first randomly selected value is a success)?
____0.55________
3c. What is P(the second randomly selected value is a failure)?
_____0.45_______
4. If our null hypothesis about the probability of a success is H0: P=0.38 then
4a. What is P(the first randomly selected value is a failure)?
______0.62______
4b. What is P(the fifth randomly selected value is a success)?
_____0.38_______
4c. What is P(that any randomly selected value is a success)?
_____0.38_______
5. Your “significant other” wants to get you a present. The present will be lessons in an
adventure activity. The three choices are SCUBA diving, Flying and Rock Climbing. The table
below shows the distribution of former statistics classes from which one individual will be
randomly selected.
Each number represents a
count for the row/column
combination.
Gender
Male
Female
Column Total
Adventure Lesson
SCUBA
Flying
Climbing
Row Total
15
29
44
13
24
37
49
83
132
21
30
51
(Update after U10)
5a. What is P(SCUBA)?
____
______
5b. What is P(Flying or Climbing)?
___
_______
5c. What is P(Male or Flying)?
____
______
5d. What is P(Female or SCUBA)?
____
______
6. You have the option of visiting a different part of the world. The choices are Europe, Asia,
South America. The table below represents the distribution of former statistics classes from
which one individual will be randomly selected.
Each number represents a
count for the row/column
combination.
Gender
Male
Female
Column Total
Travel Location
Europe
Asia
21
24
45
7
11
18
South
America
5
11
16
Row Total
33
46
79
(Update after U10)
6a. Find P(Europe).
____ _______
6b. Find P(Female)
____ _______
6c. Find P(Male or South America)
____ _______
6d. Find P(Europe or Asia)
____ _______
7. If our null hypothesis about the probability of a success is H0: P=0.6 then if S = success and F
= Failure:
7a. Find P(SF)
___0.24________
7b. Find P(FS)
____0.24_______
7c. Find P(SS)
___0.36________
7d. Find P(FF)
_____0.16______
7e. Complete the chart, where X is the random variable for the number of successes.
X=x
P(X = x)
0
0.16
1
0.48
2
0.36
7f. Make a stick graph showing the probability of 0, 1, 2 successes.
7g. What is the mean number of successes?
7h. What is the standard deviation of the number of successes?
__1.2_____
__0.6928_____
8. If our null hypothesis about the probability of a success is H0: P=0.3 then if S = success and F
= Failure and the sample size of from this population is 5:
8a. Find P(SSSSS)
____0.00243_____
8b. Find P(at least 1 failure)
8c. Find P(SSSSF)
____0.99757________
____0.00567________
8d. Find P(SFSSS)
8e. How many ways are there to have 4 successes in this sample?
____0.00567________
___5_________
8f. What is the probability for exactly 4 successes?
___0.02835_________
8g. Use your calculator to complete the table where the random variable X is the number of
successes.
X=x
0
1
2
3
4
5
P(X = x) 0.16807 0.36015 0.3087 0.1323 0.02835 0.00243
8h. Make a stick graph showing the probability for each number of successes.
8i. What is the mean number of successes?
8j. What is the standard deviation for the number of successes?
____1.5_____
______1.02_____
9. If our null hypothesis about the probability of a success is H0: P=0.65 and H1: P>0.65, and if a
sample of 250 units had 176 successes, then what is the exact probability of getting 176 or more
successes out of 250 units? At the 0.05 level of significance, which hypothesis is supported?
_____0.0411___ H1___
10. If our null hypothesis about the probability of a success is H0: P=0.25 and H1: P>0.25, and if
a sample of 80 units had 22 successes, then what is the exact probability of getting 22 or more
successes out of 80 units? At the 0.1 level of significance, which hypothesis is supported?
____0.3426__ H0___
11. If our null hypothesis about the probability of a success is H0: P=0.80 and H1: P<0.80, and if
a sample of 820 units had 642 successes, then what is the exact probability of getting 642 or less
successes out of 820 units? At the 0.05 level of significance, which hypothesis is supported?
____0.1199 H0_______
12. If our null hypothesis about the probability of a success is H0: P=0.45 and H1: P<0.45, and if
a sample of 110 units had 38 successes, then what is the exact probability of getting 38 or less
successes out of 110 units? At the 0.05 level of significance, which hypothesis is supported?
____0.0168___ H1_____
13. The standard rate for tipping a waitress/waiter is 15%. The manager of a restaurant
hypothesized that over half the people tipped more than that amount when paying by credit card.
After analyzing 350 credit card receipts, the manager found that 185 people tipped over 15%.
13a. There are two different percents or proportions shown in this problem. One is 15%, what is
the other?
______50% or 0.50__________
13b. Of the two, which is the one that will be included in the hypotheses?____0.50___________
13c. Since the data consists of two possible responses (tipped more than the standard rate, tipped
less than or equal to the standard rate) then is this problem about means or proportions?
_____proportion__________
13d. Write the appropriate null and alternate hypothesis.
H0:___p=0.5___________
H1:___p>0.5___________
13e. Using Method 1: The Binomial Distribution, determine the exact probability that 185 or
more people out of 350 would tip over 15%. Which hypothesis is supported if α = 0.05?
p-value__0.1549__, hypothesis_ H0___
13f. What is the mean and standard deviation of the binomial distribution?µ=175, σ=9.35
13g. Using Method 2: The normal approximation to the binomial distribution, draw and label
the normal curve then determine the approximate probability that 185 or more out of 350 would
tip over the standard rate. Which hypothesis is supported if α = 0.05?
p-value_0.1425 (calc)___, hypothesis_ H0__
13h. What is the sample proportion of people who tipped over the standard rate?
____
0.5286______
13i. Using Method 3: The sampling distribution of sample proportions, draw and label the
normal curve then determine the approximate probability of getting the sample proportion, or
one more extreme, from the hypothesized distribution. Which hypothesis is supported if α =
0.05?
p-value__0.1425 (calc)___, hypothesis__ H0___
14. Living sustainably is a method of living such that each person can live a worthwhile life
without jeopardizing the opportunity of future generations to also live a worthwhile life. It
includes issues of the environment, economy and social justice. It is estimated that fewer than
10% of people are familiar with living sustainably. A random sample of 560 people showed that
42 were familiar with the concept of living sustainably.
14a. Since the data consists of two possible responses (familiar, not familiar), then is this
problem about means or proportions?
_____proportions_______
14b. Write the appropriate null and alternate hypothesis.
H0:___p=0.10________
H1:__p<0.01_________
14c. Using Method 1: The Binomial Distribution, to determine the exact probability of the
sample results. Which hypothesis is supported if α = 0.05?
p-value___0.02512___, hypothesis__ H1_
14d. What is the mean and standard deviation of the binomial distribution?
Mean__µ=56__Standard Deviation_σ=7.1___
14e. Using Method 2: The normal approximation to the binomial distribution, draw and label
the normal curve then determine the approximate probability of the sample results. Which
hypothesis is supported if α = 0.05?
p-value___0.0243 (calculator)__, hypothesis__ H1__
14f. What is the sample proportion of people familiar with sustainable living?
____
0.0.075_____
14g. Using Method 3: The sampling distribution of sample proportions, draw and label the
normal curve then determine the approximate probability of getting the sample proportion, or
one more extreme, from the hypothesized distribution. Which hypothesis is supported if α =
0.05?
p-value__0.0243 (calc)__, hypothesis__H1__
15. Since moving closer to work and trying to walk instead of drive, one person wondered if his
average walking distance was over 3 miles a day. Assume the amount he walked was normally
distributed and the standard deviation of the amount he walked is 0.4 miles.
15a. Is this problem about a mean or proportion?
______mean_____
15b. If he randomly selected 16 walking days, then what is the mean of the distribution of
sample means?
3 miles
15c. If he randomly selected 16 walking days, then what is the standard deviation of the
distribution of sample means?
0.1 miles
15d. Draw and label the normal curve for the distribution of sample means.
15e. The amount he walked on randomly selected days is shown in the table below.
3.3
2.5
3.1
3.4
2.9
3.4
2.6
2.9
4.0
3.9
3.0
3.2
3.8
3.7
What is the sample mean and standard deviation?
3.4
2.6
mean_3.23 standard deviation__0.4686__
15f. For the hypotheses H0: μ = 3 and H1: μ>3, what is the probability of getting this sample
mean, or one more extreme for the sampling distribution that is based on the null hypothesis and
a sample of size 16? Which hypothesis is supported if α = 0.05?
p-value__0.0107__ hypothesis__ H1__
16. As a result of moving closer to work and walking instead of driving so much, not only did
the person from the last problem lose 20 pounds and reduced his blood pressure, he also found he
had extra money because he could pay less for his car insurance, had fewer car repairs and paid
less for gas. He hypothesized that the average savings per week in money spent on gas was over
$40. Assume the standard deviation for the amount he spent on gas before moving was $8.00
16a. Is this problem about a mean or proportion?
______mean______
16b. If he randomly selected 25 weeks from before he moved, then what is the mean of the
distribution of sample means for the amount he spent on gas?
µ=40
16c. If he randomly selected 25 weeks from before he moved, then what is the standard
deviation of the distribution of sample means for the amount he spent on gas
σ=1.60
16d. Draw and label the normal curve for the distribution of sample means.
16e. For the hypotheses H0: μ = 40 and H1: μ>40, if he found the average he spent during 25
weeks to be $42.31, what is the probability of getting this sample mean, or one more extreme for
the sampling distribution that is based on the null hypothesis and a sample of size 25?
p = 0.0744