Final_Z score exercise (1)
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Clark University *
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1150
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Mathematics
Date
Jan 9, 2024
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docx
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Uploaded by LieutenantTank10558
Exercise 1: Comparing Z-scores
Q1. Here are some summary statistics for the percentage of salt content for two seasoning
blends a company produces.
Kaylee bought a bottle of seasoning salt that contained 59% percent salt and a bottle of
Cajun seasoning that contained 44% percent salt.
Relative to the type of blend, which bottle had a lower salt content?
Choose 1 answer:
A.
Kaylee's bottle of seasoning salt
B.
Kaylee's bottle of Cajun seasoning
C.
Both of her bottles had equal salt content relative to their blends.
D.
It's impossible to say without seeing all of the individual salt contents.
E.
It's impossible to say since we don't know the shape of either distribution.
Q2. The Appalachian Trail is a long hiking trail that takes several months to travel by foot.
Hikers sometimes change sleeping bags during the trip because of changing seasons.
Each sleeping bag has a rating telling the minimum outside temperature in which it is
effective. Here are some summary statistics for the ratings for the sleeping bags hikers were
using at the beginning and end of their trip:
Irina used a sleeping bag rated 15 degrees F at the beginning of the trip and a sleeping bag
rated 29 degrees F at the end of her trip on the Appalachian Trail.
Relative to other sleeping bags hikers were using at the time, which of Irina's sleeping
bags had a relatively lower rating?
Choose 1 answer:
A.
Irina's sleeping bag at the beginning of her trip had a lower relative rating.
B.
Irina's sleeping bag at the end of her trip had a lower relative rating.
C.
Both of Irina's sleeping bags had equal relative ratings.
D.
It's impossible to say without seeing all of the individual sleeping bag ratings.
E.
It's impossible to say since we don't know the shape of either distribution.
Q3. People tend to be members in the Senate later in their careers than they are in the
House of Representatives. Here are some summary statistics for the ages of members of the
House of Representatives and the Senate one year in the United States.
The Senate minority leader that year was 66 years old, while the House of Representatives
minority leader was 76 years old.
Relative to their group, which leader was older?
Choose 1 answer:
A.
The Senate minority leader
B.
The House of Representatives minority leader
C.
The leaders had equal ages relative to their groups.
D.
It's impossible to say without seeing all of the individual ages.
E.
It's impossible to say since we don't know the shape of either distribution.
Q4. Researchers in an early childhood development organization investigated the number of
words in the vocabularies of young children. Here are some summary statistics:
Miku (who is 18 months old) has a vocabulary of 93 words, while Tessa (who is 24 months
old) has a vocabulary of 287 words.
Relative to their age, who has a larger vocabulary?
Choose 1 answer:
A.
Miku
B.
Tessa
C.
Miku and Tessa have equally sized vocabularies relative to their ages.
D.
It’s impossible to say without seeing all of the individual vocabulary sizes.
E.
It’s impossible to say since we don’t know the shape of either distribution.
Exercise 2: Calculating z-scores
1.
The grades on a history midterm at Gardner Bullis are roughly symmetric
with μ=69 and σ=3.5. Ishaan scored 65 on the exam.
Find the z-score for Ishaan's exam
grade. Round to two decimal places.
Z score = -1.14
2.
The grades on a language midterm at Santa Rita are roughly symmetric with μ=77
and σ=3.5. Tiffany scored 80 on the exam.
Find the z-score for Tiffany's exam grade.
Round to two decimal places.
Z score = 0.86
3.
The grades on a history midterm at Almond are roughly symmetric with μ=85 and σ=2.0.
Michael scored 86 on the exam.
Find the z-score for Michael's exam grade. Round to
two decimal places.
-
Z score = 0.50
4.
The grades on a geometry midterm at Oak are roughly symmetric with μ=74 and σ=4.0.
Nadia scored 70 on the exam.
Find the z-score for Nadia's exam grade. Round to two
decimal places.
-
Z score = -1.00
- WORK SHOWN ON FIFTH PAGE
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Exercise 3: Calculating z-scores
PROBLEM 1
Nutritionists measured the sugar content (in grams) for 32 drinks at Jake's Java coffee shop. The
drinks had a mean of 18 g and a standard deviation of 5 g and the distribution was roughly
symmetric. A Grande Mocha Cappuccino at Jake's Java contains 14 g of sugar.
Calculate the standardized score (z-score) for the Grande Mocha Cappuccino.
Z =
-0.8
PROBLEM 2
What is the best interpretation of the z-score from the previous problem?
A: This drink has 0.8 g of sugar less than the mean of the 32 drinks.
B:
This drink is 0.8 standard deviations below the mean of the 32 drinks.
C: About 80% of the drinks have a lower sugar content than this drink.
D: About 80% of the drinks have a higher sugar context than this drink.
PROBLEM 3
A different coffee shop called Ruth's Roasts serves drinks that contain a mean of 16 g sugar with
a standard deviation of 6g.
A Grande Mocha Cappuccino at Ruth's Roasts contains 13g of sugar.
Assuming that the distributions of sugar amounts have approximately the same shape, at
which shop does this drink have less sugar relative to the other drinks at its respective store?
A: This drink has relatively less sugar at Jake’s Java.
B: This drink has relatively less sugar at Ruth’s Roast.
C: The relative sugar contents are equal at both locations.
D:
It is impossible to tell because we do not know how many drinks are offered at Ruth's Roasts.
CHALLENGE PROBLEM
Drinks at Ben's Beans have a mean sugar content of 20g with a standard deviation of 6g. The
sugar content in a Grande Cappuccino at Ben's Beans has a z-score of −0.75 compared to the
rest of the drinks at that shop.
How much sugar is in a Grande Cappuccino at Ben's Beans?
15.5 grams
WORK SHOWN ON FIFTH PAGE