Statistics Assignment: Hypothetical Scenarios on Z-score, Confidence Intervals, and Sampling

Module 3 Assignment
Chapter 5
1. (Hypothetical) A random sample of 600 daily newspapers found that on average,
each edition contained 3.0local crime stories per day, with a standard deviation of
2.0. Answer the following questions assuming that the distribution of crime stories
in newspapers is normal. (25 points possible)
a) What is the Z score associated with a newspaper with 6 local crime stories?
Z = (X - μ) / σ
Z = (6 - 3) / 2
Z = 1.5
b) What proportion of newspapers published 2 or fewer crime stories? How many
does this correspond to in the sample? (Round your answer to nearest integer.)
Z = (X - μ) / σ
Z = (2 - 3) / 2
Z = -0.5
c) What number of news stories corresponds to a Z score of +2? (Round your
answer to nearest integer.)
Z = (X - μ) / σ
2= (x- 3) / 2
x = 7
7 news stories.
d) What percentage of newspapers publish between 1 and 6 local crime stories per
day?
Proportion of newspapers published six or fewer crime stories:
Z score= (Value – Mean) / Standard Deviation
Z score = (6.00-3) / 2.0
Z score = 1.5

Using the normal distribution tables, the percentage of newspapers that publish between 1 and 6
stories correspond to 0.9772 i.e. 97.72% of the papers
Hence of all the newspapers, 97.72*600≈ 586 newspapers publish between 1 and 6 stories.
Chapter 6
2. Can the standard error of the mean ever be larger than, or even equal to, the standard
deviation for the same variable? Justify your answer by means of both a formula and a
discussion of the relationship between these two concepts. (5 points possible)
Answer
In the event that the sample size gets smaller, the standard error of the sample will get bigger
since the standard error of the mean aims to tell us how close to the estimator the population
parameter is. As such, the sample mean is inversely proportional to the mean.
Proof
Since the standard error of the mean implies the standard deviation of the distribution from of
sample means taken from a population. The smaller the standard error, the more representative
the sample will be of the overall population.
Formula:
Where σM is the standard error of the mean. From the above formula, the
smaller the sample size, the larger the S.E.
Chapter 7
3. (40 points possible.) (Hypothetical) You are conducting research on the prevalence of
severe binge drinking among college students. You define severe binge drinking as
consuming 5 or more alcoholic beverages at a single sitting. You ask the question: “On
how many days in the past two weeks have you consumed 5 or more drinks at one time?
The answers can range from 0 to 14. You collect data on a random sample of 400
students.
The average for this sample in 1.42 with a standard deviation of .80. Construct a 95%
confidence interval for the average binge drinking score in the population.
Solution
Since the sample size is greater than 30, we use a Z-test:
μ = M ± Z(sM)
Where:
M = sample mean
Z = Z statistic determined by confidence level
sM = standard error = √(s2/n)
Hence using the above formula,
Calculation
M = 142
Z = 1.96