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WorksheetsElementary Statistics & Probability Final Examination
Total questions: 50
Worksheet time: 1hrs 15mins
Which of the following best defines probability?
The number of favorable outcomes
The measure of the likelihood that an event will occur
The total number of possible outcomes
The ratio of successful to unsuccessful trials
In the classical approach to probability, the probability of an event E is calculated as:
P(E) = Number of favorable outcomes / Total number of outcomes
P(E) = Number of trials / Number of successes
P(E) = Frequency / Total observations
P(E) = Total outcomes - Unfavorable outcomes
The empirical approach to probability is based on:
Theoretical calculations
Personal judgment and experience
Historical data and observed frequencies
Mathematical formulas only
If you roll a fair six-sided die, what is the probability of getting a number greater than 4?
1/6
1/3
1/2
2/3
Two events A and B are mutually exclusive. If P(A) = 0.3 and P(B) = 0.4, what is P(A or B)?
0.12
0.70
0.10
1.00
The Addition Rule for non-mutually exclusive events states that:
P(A or B) = P(A) + P(B)
P(A or B) = P(A) + P(B) - P(A and B)
P(A or B) = P(A) × P(B)
P(A or B) = P(A) - P(B)
In a deck of 52 cards, what is the probability of drawing either a King or a Heart?
4/13
17/52
16/52
1/4
Conditional probability P(A|B) represents:
The probability of A and B occurring together
The probability of A given that B has occurred
The probability of B given that A has occurred
The probability of either A or B occurring
A box contains 5 red balls and 3 blue balls. If you draw one ball, then draw another WITHOUT replacement, what is the probability of drawing two red balls?
25/64
5/14
20/56
5/16
The Multiplication Rule for independent events A and B states that:
P(A and B) = P(A) + P(B)
P(A and B) = P(A) × P(B)
P(A and B) = P(A) × P(B|A)
Both B and C are correct
If P(A) = 0.6, P(B) = 0.5, and events A and B are independent, what is P(A and B)?
1.1
0.3
0.55
0.1
A bag contains 4 red marbles, 3 blue marbles, and 3 green marbles. What is the probability of NOT selecting a blue marble?
3/10
7/10
4/10
1/3
In a survey, 40% of people like coffee, 30% like tea, and 15% like both. What is the probability that a randomly selected person likes coffee or tea?
0.70
0.55
0.85
0.12
The normal distribution is characterized by which of the following properties?
It is symmetrical and bell-shaped
It has a skewed distribution
It has two modes
It is uniform in shape
In a normal distribution, the mean, median, and mode are:
Different values
Equal to each other
Cannot be determined
Always equal to zero
According to the Empirical Rule, approximately what percentage of data falls within one standard deviation of the mean in a normal distribution?
50%
68%
95%
99.7%
According to the Empirical Rule, approximately 95% of the data in a normal distribution falls within:
±1 standard deviation from the mean
±2 standard deviations from the mean
±3 standard deviations from the mean
±4 standard deviations from the mean
The standard score (z-score) represents:
The raw score in the distribution
The number of standard deviations a value is from the mean
The mean of the distribution
The standard deviation of the distribution
If a student's test score has a z-score of -1.5, this means:
The score is 1.5 points below the mean
The score is 1.5 standard deviations below the mean
The score is 1.5 standard deviations above the mean
The score is average
PROBLEM SOLVING: The mean score on a statistic exam is 75 with a standard deviation of 8. What is the z-score for a student who scored 83?
(a)
In a standard normal distribution, the mean is _____ and the standard deviation is ___.
0, 1
1, 0
0, 0
1, 1
If z = 1.96, approximately what percentage of data lies below this z-score in a standard normal distribution?
47.5%
95%
97.5%
99%
PROBLEM SOLVING: The heights of the students in a class are normally distributed with a mean of 165 cm and standard deviation of 10 cm? What percentage of students have heights between 155 cm and 175 cm? (Use the Empirical Rule)
(a)
The null hypothesis (Ho) is:
The hypothesis we want to prove
A statement of no effect or no difference
Always accepted in hypothesis testing
The alternative to the research hypothesis
The alternative hypothesis (H1 or Ha) represents:
The hypothesis of no difference
The researcher's claim or what we want to support
The null hypothesis
The accepted hypothesis
A Type I error occurs when:
We fail to reject a true null hypothesis
We reject a true null hypothesis
We reject a false null hypothesis
We accept the alternative hypothesis correctly
The level of significance (α) commonly used in hypothesis testing is:
0.01 or 0.05
0.50
1.00
0.95
A two-tailed test is used when:
We test if a parameter is greater than a value
We test if a parameter is less than a value
We test if a parameter is different from a value (either greater or less)
We already know the direction of the difference
Which of the following is the correct sequence of steps in hypothesis testing?
State hypotheses → Set significance level → Collect data → Compute test statistic → Make decision
Collect data → State hypotheses → Compute test statistic → Set significance level → Make decision
Set significance level → Collect data → State hypotheses → Compute test statistic → Make decision
Compute test statistic → State hypotheses → Set significance level → Make decision
The p-value in hypothesis testing represents:
The probability that the null hypothesis is true
The probability of obtaining results as extreme as observed, assuming H0 is true
The probability that the alternative hypothesis is true
The level of significance
If the p-value is less than the significance level (α), we:
Fail to reject the null hypothesis
Reject the null hypothesis
Accept the null hypothesis
Reject the alternative hypothesis
The z-test is appropriate when:
The sample size is small (n < 30) and population standard deviation is unknown
The sample size is large (n ≥ 30) or population standard deviation is known
Testing the relationship between two variables
Comparing three or more groups
The t-test is used instead of the z-test when:
The population standard deviation is known
The sample size is large
The sample size is small and population standard deviation is unknown
Testing correlation
PROBLEM SOLVING: A researcher claims that the average height of students is 170 cm. A sample of 40 students has a mean height of 168 cm with a standard deviation of 6 cm. Calculate the z-test statistic. (Round to 2 decimal places)
(a)
ANOVA (Analysis of Variance) is used to:
Compare means of two groups
Compare means of three or more groups
Test correlation between two variables
Determine the standard deviation
In ANOVA, if the F-statistic is large and the p-value is small, we conclude:
There is no significant difference among group means
There is a significant difference among at least two group means
All groups have equal means
The null hypothesis is true
Correlation analysis measures:
The cause-and-effect relationship between variables
The strength and direction of the linear relationship between variables
The difference between two variables
The variance within a variable
The correlation coefficient (r) ranges from:
0 to 1
-1 to 0
-1 to +1
0 to infinity
A correlation coefficient of r = -0.85 indicates:
Strong positive correlation
Weak positive correlation
Strong negative correlation
No correlation
If r = 0, this indicates:
Perfect positive correlation
Perfect negative correlation
No linear correlation
Strong correlation
Pearson's Product Moment Correlation Coefficient is used when:
Both variables are ordinal/ranked data
Both variables are continuous and normally distributed
Variables are categorical
One variable is nominal
The coefficient of determination (r²) represents:
The correlation coefficient squared
The percentage of variance in one variable explained by the other
The strength of the relationship
Both A and B are correct
PROBLEM SOLVING: If the correlation coefficient between study hours and exam scores is r = 0.80, what is the coefficient of determination (r²)? Express as a percentage.
(a)
A positive correlation means that:
As one variable increases, the other decreases
As one variable increases, the other also increases
There is no relationship between variables
The variables are independent
Spearman's Rank Order Correlation is used when:
Data is continuous and normally distributed
Data is ordinal or ranked
Data is nominal
Testing differences between groups
Which value of correlation coefficient indicates the strongest relationship?
r = 0.45
r = -0.72
r = 0.30
r = -0.15
The Chi-Square test is used to:
Measure the strength of correlation between continuous variables
Test the association between categorical variables
Compare means of groups
Calculate the correlation coefficient
In Chi-Square analysis, we compare:
Observed frequencies with expected frequencies
Means with standard deviations
Correlation coefficients
Z-scores with t-scores
PROBLEM SOLVING: The correlation between advertising expenditure (in thousands) and sales (in thousands) for 5 products is being studied. Given: Σx = 50, Σy = 100, Σxy = 1150, Σx² = 550, Σy² = 2100, n = 5. Calculate Pearson's r. (Round to 2 decimal places)
(a)
If the Chi-Square calculated value is greater than the Chi-Square critical value, we:
Fail to reject the null hypothesis
Reject the null hypothesis and conclude there is an association
Accept the null hypothesis
Conclude there is no relationship
