1What is the primary purpose of the Null Hypothesis () in statistical testing?
A.To calculate the standard deviation of the population
B.To state that there is no significant difference or relationship
C.To determine the sample size required
D.To prove that a relationship exists between variables
Correct Answer: To state that there is no significant difference or relationship
Explanation:
The Null Hypothesis () proposes that there is no statistical significance exists in a set of given observations. It is the hypothesis that the researcher tries to disprove or reject.
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2Which of the following errors occurs when the Null Hypothesis is rejected when it is actually true?
A.Type II Error
B.Type I Error
C.Standard Error
D.Sampling Error
Correct Answer: Type I Error
Explanation:
A Type I Error (False Positive) occurs when the null hypothesis () is true, but we incorrectly reject it. The probability of this error is denoted by .
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3The probability of committing a Type II Error is denoted by which Greek letter?
A. (Chi)
B. (Beta)
C. (Sigma)
D. (Alpha)
Correct Answer: (Beta)
Explanation:
Type II Error (failing to reject a false null hypothesis) is denoted by the symbol . The power of the test is calculated as .
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4What does the Power of a Test represent?
A.The probability of accepting a true Null Hypothesis
B.The probability of making a Type I error
C.The probability of correctly rejecting a false Null Hypothesis
D.The probability of making a Type II error
Correct Answer: The probability of correctly rejecting a false Null Hypothesis
Explanation:
The Power of a Test () is the probability that the test correctly rejects the Null Hypothesis () when the Alternative Hypothesis () is true.
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5Which of the following conditions implies a Two-tailed Test?
A.
B.
C.
D.
Correct Answer:
Explanation:
A two-tailed test is used when the alternative hypothesis specifies that the parameter is not equal to the null value, meaning the difference could be in either direction (positive or negative).
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6In a hypothesis test, if the p-value is less than the level of significance (), what is the conclusion?
A.Accept the Null Hypothesis
B.The test is inconclusive
C.Change the level of significance
D.Reject the Null Hypothesis
Correct Answer: Reject the Null Hypothesis
Explanation:
If , the observed result is statistically significant, providing strong evidence against the null hypothesis, leading to its rejection.
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7Which of the following is a fundamental assumption of Parametric Tests?
A.The sample size must be small
B.The data follows a normal distribution
C.No assumption about population parameters is made
D.The data is always qualitative
Correct Answer: The data follows a normal distribution
Explanation:
Parametric tests (like t-tests and Z-tests) assume that the data is derived from a population that follows a specific probability distribution, typically the Normal Distribution.
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8The Z-test is generally used when the sample size () is:
A.
B.
C.
D.
Correct Answer:
Explanation:
The Z-test is appropriate for large sample sizes, typically defined as . For smaller samples where population variance is unknown, the t-test is preferred.
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9What is the formula for the Z-statistic when testing a sample mean against a population mean?
A.
B.
C.
D.
Correct Answer:
Explanation:
The Z-statistic is calculated by taking the difference between the sample mean and population mean, divided by the standard error of the mean ().
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10Student's t-test was developed by whom?
A.R.A. Fisher
B.Karl Pearson
C.Wilcoxon
D.William Sealy Gosset
Correct Answer: William Sealy Gosset
Explanation:
The t-test was developed by William Sealy Gosset under the pseudonym 'Student'.
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11What is the shape of the t-distribution compared to the standard normal distribution?
A.It is positively skewed
B.It is negatively skewed
C.It is flatter and has thicker tails (leptokurtic)
D.It is identical to the normal distribution
Correct Answer: It is flatter and has thicker tails (leptokurtic)
Explanation:
The t-distribution is symmetric and bell-shaped like the normal distribution, but it has heavier tails (is flatter) to account for the greater uncertainty associated with small sample sizes.
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12Degrees of freedom () for a single sample t-test is calculated as:
A.
B.
C.
D.
Correct Answer:
Explanation:
For a one-sample t-test, the degrees of freedom are calculated as the sample size minus one ().
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13Which test is most appropriate to compare the means of two independent small samples () taken from normal populations with unknown variances?
A.F-test
B.Chi-Square test
C.Independent samples t-test
D.Z-test
Correct Answer: Independent samples t-test
Explanation:
The independent samples t-test is designed to compare means between two unrelated groups when sample sizes are small and population variance is unknown.
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14In a Paired t-test, the samples are:
A.Related or matched pairs
B.Categorical data only
C.Always larger than 30
D.Independent of each other
Correct Answer: Related or matched pairs
Explanation:
A paired t-test is used when data points in the two groups are related, such as 'before and after' measurements on the same subjects.
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15The F-test is primarily used to test the equality of:
A.Two Medians
B.Two Means
C.Two Variances
D.Two Proportions
Correct Answer: Two Variances
Explanation:
The F-test utilizes the F-distribution to compare two population variances ( and ) to determine if they are significantly different.
The F-statistic is calculated as , where is the larger variance and is the smaller variance.
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17The Chi-Square () test is a:
A.Test for Means
B.Test for Standard Deviation
C.Parametric test
D.Non-parametric test
Correct Answer: Non-parametric test
Explanation:
The Chi-Square test is a non-parametric test because it does not assume a normal distribution of the population and is often used for categorical data.
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18What is the formula for the Chi-Square statistic?
A.
B.
C.
D.
Correct Answer:
Explanation:
The Chi-Square statistic is the sum of the squared differences between Observed () and Expected () frequencies, divided by the Expected frequency.
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19In a Chi-Square test of independence for a contingency table with rows and columns, the degrees of freedom is:
A.
B.
C.
D.
Correct Answer:
Explanation:
For a contingency table, the degrees of freedom are calculated as the product of (rows minus 1) and (columns minus 1).
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20The Kruskal-Wallis test is a non-parametric alternative to which parametric test?
A.Z-test
B.Independent t-test
C.Paired t-test
D.One-way ANOVA
Correct Answer: One-way ANOVA
Explanation:
The Kruskal-Wallis test is used to compare more than two independent groups when the assumptions of ANOVA (like normality) are not met.
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21What type of data is primarily used in the Kruskal-Wallis test?
A.Ranked (Ordinal) data
B.Ratio data
C.Nominal data only
D.Interval data
Correct Answer: Ranked (Ordinal) data
Explanation:
The Kruskal-Wallis test is a rank-based test. It converts continuous or ordinal data into ranks to compute the test statistic.
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22Which of the following values represents the critical Z-score for a 95% Confidence Level (Two-tailed)?
A.2.33
B.2.58
C.1.96
D.1.645
Correct Answer: 1.96
Explanation:
For a 95% confidence level in a two-tailed test, the significance level is 0.05, split into 0.025 on each tail. The Z-score corresponding to this is 1.96.
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23In hypothesis testing, the Significance Level () is typically set at:
A.0.05 or 0.01
B.0.50 or 0.10
C.1.00
D.0.95 or 0.99
Correct Answer: 0.05 or 0.01
Explanation:
The standard levels of significance used in research are 0.05 (5%) and 0.01 (1%).
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24Which test statistic follows a distribution that is strictly non-negative (from 0 to positive infinity)?
A.Z-distribution
B.Chi-Square distribution
C.Normal distribution
D.t-distribution
Correct Answer: Chi-Square distribution
Explanation:
Because the Chi-Square statistic is calculated by summing squared quantities, it can never be negative. It ranges from 0 to .
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25When testing for Goodness of Fit, which test is applied?
A.Z-test
B.F-test
C.Chi-Square test
D.t-test
Correct Answer: Chi-Square test
Explanation:
The Chi-Square Goodness of Fit test determines if a sample data matches a population with a specific distribution.
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26If the calculated value of a test statistic is greater than the critical table value, we should:
A.Accept
B.Reject
C.Retest with more data
D.Reject
Correct Answer: Reject
Explanation:
When the test statistic falls into the critical region (value > critical value), it implies the result is statistically significant, so we reject the Null Hypothesis.
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27For a One-Way ANOVA (Analysis of Variance), what is the appropriate test statistic?
A.t-statistic
B.Z-statistic
C.F-statistic
D.-statistic
Correct Answer: F-statistic
Explanation:
ANOVA uses the F-test to determine if there are any statistically significant differences between the means of three or more independent (unrelated) groups.
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28In the Kruskal-Wallis test, the test statistic is usually denoted by:
A.W
B.K
C.H
D.U
Correct Answer: H
Explanation:
The Kruskal-Wallis test statistic is standardly denoted as .
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29Standard Error (SE) of the mean decreases when:
A.The sample size decreases
B.The sample mean increases
C.The population standard deviation increases
D.The sample size increases
Correct Answer: The sample size increases
Explanation:
Since , increasing the sample size () in the denominator reduces the Standard Error.
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30Which assumption is NOT required for the Chi-Square Test?
A.Observations must be independent
B.Data must be in the form of frequencies
C.Population must be normally distributed
D.Expected frequency in any cell should not be less than 5
Correct Answer: Population must be normally distributed
Explanation:
The Chi-Square test is non-parametric and does not assume a normal distribution of the population.
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31When , the Student’s t-distribution approaches which distribution?
A.F-distribution
B.Standard Normal (Z) distribution
C.Chi-Square distribution
D.Binomial distribution
Correct Answer: Standard Normal (Z) distribution
Explanation:
As the sample size (and consequently degrees of freedom) increases, the t-distribution becomes narrower and approaches the standard normal Z-distribution.
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32In an F-test, if and , what is the calculated F-value?
A.5.0
B.4.0
C.75
D.0.25
Correct Answer: 4.0
Explanation:
.
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33A researcher wants to test if the variance of Machine A is significantly different from Machine B. Which test should be used?
A.F-test
B.Chi-Square test
C.Z-test
D.t-test
Correct Answer: F-test
Explanation:
The F-test is specifically designed to compare the variances of two populations.
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34The degrees of freedom for the Kruskal-Wallis test with groups is:
A.
B.
C.
D.
Correct Answer:
Explanation:
The degrees of freedom for the Kruskal-Wallis statistic is , where is the number of groups being compared.
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35Which of the following is true regarding Non-parametric tests?
A.They are more powerful than parametric tests when assumptions are met
B.They can only be used for small samples
C.They are also known as distribution-free tests
D.They rely on strict assumptions about population parameters
Correct Answer: They are also known as distribution-free tests
Explanation:
Non-parametric tests do not assume a specific distribution (like the normal curve) for the population, hence the name distribution-free.
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36What is the Alternative Hypothesis () for a Chi-Square test of independence?
A.The variables are not independent (there is an association)
B.The means are equal
C.The variables are independent
D.The variances are equal
Correct Answer: The variables are not independent (there is an association)
Explanation:
In a Chi-Square test of independence, assumes independence. Therefore, states that the variables are associated (not independent).
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37The Mann-Whitney U Test is the non-parametric equivalent of which parametric test?
A.Pearson Correlation
B.Paired t-test
C.One-way ANOVA
D.Independent samples t-test
Correct Answer: Independent samples t-test
Explanation:
The Mann-Whitney U test compares differences between two independent groups when the dependent variable is ordinal or continuous but not normally distributed, similar to the independent t-test.
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38If the Standard Deviation of a population is unknown and , we calculate the standard error using:
A.Population standard deviation ()
B.Sample standard deviation ()
C.Variance
D.Range
Correct Answer: Sample standard deviation ()
Explanation:
When is unknown, it is estimated by the sample standard deviation , necessitating the use of the t-distribution.
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39Which of the following symbols represents the sample mean?
A.
B.
C.
D.
Correct Answer:
Explanation:
(x-bar) represents the sample mean, while (mu) represents the population mean.
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40In an F-test, the degrees of freedom are denoted as . What does represent?
A.Degrees of freedom of the numerator (larger variance)
B.Degrees of freedom of the denominator (smaller variance)
C.Number of groups
D.Total sample size
Correct Answer: Degrees of freedom of the numerator (larger variance)
Explanation:
The F-distribution is defined by two degrees of freedom. corresponds to the variance in the numerator (), and corresponds to the variance in the denominator ().
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41What is the minimum expected frequency typically recommended for each cell in a Chi-Square contingency table?
A.30
B.1
C.10
D.5
Correct Answer: 5
Explanation:
A common rule of thumb for the Chi-Square test is that the expected frequency in each cell should be at least 5 to ensure the approximation to the Chi-Square distribution is valid.
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42A Type I Error is also known as:
A.Confidence Level
B.Power of the Test
C.False Negative
D.False Positive
Correct Answer: False Positive
Explanation:
Rejecting a true Null Hypothesis is detecting an effect that is not actually there, hence 'False Positive'.
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43The relationship between the t-statistic and F-statistic (when comparing two groups) is:
A.
B.
C.
D.
Correct Answer:
Explanation:
When comparing two groups, the F-statistic from a one-way ANOVA is equal to the square of the t-statistic from a t-test ().
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44Which of the following is NOT a step in Hypothesis Testing?
A.Formulating and
B.Selecting the level of significance
C.Eliminating outliers before data collection
D.Calculating the test statistic
Correct Answer: Eliminating outliers before data collection
Explanation:
Eliminating outliers is part of data cleaning, but strictly speaking, the formal steps of hypothesis testing begin with formulating hypotheses and end with drawing a conclusion based on the statistic.
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45The critical region is:
A.The total area under the probability curve
B.The area under the curve where is rejected
C.The area under the curve where is accepted
D.The value of the test statistic
Correct Answer: The area under the curve where is rejected
Explanation:
The critical region (or rejection region) corresponds to the values of the test statistic that are unlikely to occur if the null hypothesis is true, leading to rejection.
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46What happens to the critical region in a Two-tailed Z-test at ?
A.It is split equally between both tails (0.025 each)
B.It is all on the left side (0.05)
C.It is disregarded
D.It is all on the right side (0.05)
Correct Answer: It is split equally between both tails (0.025 each)
Explanation:
In a two-tailed test, the rejection region is divided between the two extremes of the distribution. For 0.05, each tail contains 0.025.
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47For a Z-test of proportions, the standard error is calculated as:
A.
B.
C.
D.
Correct Answer:
Explanation:
The standard error for a proportion is or where .
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48If we want to test whether a die is unbiased (fair), which test is best?
A.Wilcoxon Rank Sum
B.Chi-Square Goodness of Fit
C.Student's t-test
D.F-test
Correct Answer: Chi-Square Goodness of Fit
Explanation:
Testing a die involves comparing observed frequencies of rolls (1 through 6) against expected uniform frequencies, which is a classic Chi-Square Goodness of Fit problem.
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49Which test would you use to analyze the relationship between Gender (Male/Female) and Voting Preference (Party A/Party B/Party C)?
A.Pearson Correlation
B.Z-test
C.t-test
D.Chi-Square Test of Independence
Correct Answer: Chi-Square Test of Independence
Explanation:
Both variables (Gender and Voting Preference) are categorical. The Chi-Square Test of Independence is used to see if there is an association between two categorical variables.
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50In the context of the Kruskal-Wallis test, if the Null Hypothesis is true:
A.The mean ranks of the groups are significantly different
B.The variances are unequal
C.The data is normally distributed
D.The mean ranks of the groups are roughly equal
Correct Answer: The mean ranks of the groups are roughly equal
Explanation:
The Null Hypothesis in Kruskal-Wallis assumes that the population distributions are identical. Therefore, the average ranks across the groups should be approximately the same.