Topics for Final Exam

1. Fundamental Rules of probability: Computing the probability of two or more events,

2. Sampling with or without replacement.

3. Discrete random variables, (expectation, variance, standard dviation Probability mass fun-
tions), Discrete Probability distributions:( Binomial, Poisson, geometric)

4. Continuous Random Variables (expectation, variance, standard deviation, probability dis-

5. Descriptive Statistics: Measures of central tendency: mean, median, mode SD, range His-
tograms, Identifying Outliers in a dataset.

6. Sampling distribution, sum of normals, Central Limit Theorem, error of estimation and
sample size.

7. Confidence interval for the mean(known and unknown) , confidence interval for a proportion.

8. Hypothesis tests: formulating null and alternative hypothesis (one or double sided), obtain-
ing p-value of a test. paired sample tests. Hypothesis Testing for a Mean (Known and/or
Unknown Variance), Hypothesis Testing for a Proportion

9. Type I and Type II error.

10. Linear regression: Fitting a linear model, computing estimates of σ 2 .

11. Know how to read normal, Student’s t, and binomial distribution tables.

These are some selected questions from the exercises that have no solution. We will be using these
for the review session in class tomorrow.

Practice Questions
1. Consider a set of n = 20 paired observations (xi, yi), where x abd y are variables. We have
the following information:
20
X 20
X 20
X 20
X 20
X
xi = 23.92, yi = 1843.21, x2i = 29.29, xi yi = 2214.66, yi2 = 170044.5
i=1 i=1 i=1 i=1 i=1

• What is the correlation between x and y?

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• Find the least square estimators β0 and β1 of the corresponding line of best fit.

• Find an estimate for σ 2 .

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2. (Q41) Twenty-four girls in Grades 9 and 10 are put on a training program. Their time
for a 40-yard dash is recorded before and after participating in a training program. The
differences between the before-training time and the after-training time for those 24 girls
are measured, so that positive difference values represent improvement in the 40-yard dash
time. Suppose that the values of those differences follow a normal distribution and they
have a sample mean 0.079 min and a sample standard deviation 0.255 min. We conduct a
statistical test to check whether this training program can reduce the mean finish time of
40-yard dash. What is the range of p-value for this test?
Some additional questions that can follow from this information:

• Identify an appropriate test for this scenario and write down the hypotheses

• What is the observed value of the test statistic of the corresponding hypotheses.

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 • Find an interval containing the p-value for the test.

• Does the training program help improve the training performance?

3. (Q16) Question 36 is another good one for practice. Cloud seeding has been studied
for many decades as a weather modification procedure. The rainfall (in acre-feet) from 20
clouds that were selected at random and seeded with silver nitrate are recorded below. The

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data are presented in increasing order:

18 18.8 19.8 21.2 21.8 22.3 23.4

(a) Compute the sample mean and the sample standard deviation.

(b) Find the first, second (median), and third sample quartiles.

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 (c) Are there any outliers in this dataset? Explain your reasoning.

4. It takes a Christmas tree about 10 years to grow from seed to a size ready for cutting. We
want to estimate the average height, µ, of a 4-year-old Christmas tree grown from a seed.
Assume that the height of a 4-year-old tree is normally distributed.
A sample of 20 trees has a mean height of 25.25 cm and a sample standard deviation of
4.5 cm. This sample produces a confidence interval for µ with a total length of 2.673 cm.
Determine the confidence level of this confidence interval.

5. Among 2046 cars made by Company A in 1999, 56 had a problem in the brake system.
Suppose that one wants to know whether the brake system defective rate for this type of
car is less that 4%. Formulate the null and alternative hypotheses and compute the p-value
for the test.

6. A company produces orange juice bottles with a volume of approximately 2 litres each.
One machine fills half of each bottle with concentrate, and another machine fills the other
half with water. Assume the two machines work independently. The volume (in litres) of
concentrate poured by the first machine follows a normal distribution with mean µC = 0.98
and variance σC2 = 0.0009. The volume of water poured by the second machine follows a
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normal distribution with mean µW = 1.02 and variance σW = 0.0016. A bottle of orange
juice is therefore a mixture of water and concentrate.
Question: What is the probability that a bottle contains more than 1.98 litres of juice?

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7.