MCQs Correlation and Regression Analysis 6

The post is about MCQs correlation and Regression Analysis with Answers. There are 20 multiple-choice questions covering topics related to correlation and regression analysis, coefficient of determination, testing of correlation and regression coefficient, Interpretation of regression coefficients, and the method of least squares, etc. Let us start with MCQS Correlation and Regression Analysis with answers.

Online Multiple-Choice Questions about Correlation and Regression Analysis with Answers

1. The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho=0$ against the one-sided alternative $H_1:\rho<0$ at a significance level of 0.193, the p-value is

 
 
 
 

2. The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho=0$ against the one-sided alternative $H_1:\rho>0$. To test $H_0:\rho =04 against the two-sided alternative $H_1:\rho\ne 0$ at a significance level of 0.193, the p-value is

 
 
 
 

3. Which one of the following statements is true?

 
 
 
 

4. The correlation coefficient

 
 
 
 

5. The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho =0$ against the one-sided alternative $H_1:\rho >0$ at a significance level of 0.193, the p-value is

 
 
 
 

6. Assuming a linear relationship between $X$ and $Y$ if the coefficient of correlation equals $-0.30$

 
 
 
 

7. What do we mean when a simple linear regression model is “statistically” useful?

 
 
 
 

8. In a simple linear regression problem, $r$ and $\beta_1$

 
 
 
 

9. The estimated regression line relating the market value of a person’s stock portfolio to his annual income is $Y=5000+0.10X$. This means that each additional rupee of income will increase the stock portfolio by

 
 
 
 

10. If the coefficient of determination is 0.49, the correlation coefficient may be

 
 
 
 

11. If the correlation coefficient ($r=1.00$) then

 
 
 
 

12. The $Y$ intercept ($b_0$) represents the

 
 
 
 

13. Which one of the following situations is inconsistent?

 
 
 
 

14. The slope ($b_1$) represents

 
 
 
 

15. Testing for the existence of correlation is equivalent to

 
 
 
 

16. Which of the following does the least squares method minimize?

 
 
 
 

17. If the correlation coefficient $r=1.00$ then

 
 
 
 

18. The true correlation coefficient $\rho$ will be zero only if

 
 
 
 

19. If you wanted to find out if alcohol consumption (measured in fluid oz.) and grade point average on a 4-point scale are linearly related, you would perform a

 
 
 
 

20. The strength of the linear relationship between two numerical variables may be measured by the

 
 
 
 

MCQs Correlation and Regression Analysis with Answers

MCQs Correlation and Regression Analysis

  • The $Y$ intercept ($b_0$) represents the
  • The slope ($b_1$) represents
  • Which of the following does the least squares method minimize?
  • What do we mean when a simple linear regression model is “statistically” useful?
  • If the correlation coefficient $r=1.00$ then
  • If the correlation coefficient ($r=1.00$) then
  • Assuming a linear relationship between $X$ and $Y$ if the coefficient of correlation equals $-0.30$
  • Testing for the existence of correlation is equivalent to
  • The strength of the linear relationship between two numerical variables may be measured by the
  • In a simple linear regression problem, $r$ and $\beta_1$
  • The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho=0$ against the one-sided alternative $H_1:\rho<0$ at a significance level of 0.193, the p-value is The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho =0$ against the one-sided alternative $H_1:\rho >0$ at a significance level of 0.193, the p-value is
  • The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho=0$ against the one-sided alternative $H_1:\rho>0$. To test $H_0:\rho =04 against the two-sided alternative $H_1:\rho\ne 0$ at a significance level of 0.193, the p-value is
  • If you wanted to find out if alcohol consumption (measured in fluid oz.) and grade point average on a 4-point scale are linearly related, you would perform a
  • The correlation coefficient
  • If the coefficient of determination is 0.49, the correlation coefficient may be
  • The estimated regression line relating the market value of a person’s stock portfolio to his annual income is $Y=5000+0.10X$. This means that each additional rupee of income will increase the stock portfolio by
  • Which one of the following situations is inconsistent?
  • Which one of the following statements is true?
  • The true correlation coefficient $\rho$ will be zero only if
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MCQs Probability Quiz

The post is about the Online MCQs Probability Quiz. There are 20 multiple-choice questions covering topics related to random experiments, random variables, expectations, rules of probability, events and types of events, and sample space. Let us start with the Probability Quiz.

Please go to MCQs Probability Quiz to view the test

MCQs Probability Quiz

Online MCQs Probability Quiz with Answers

  • Consider a dice with the property that the probability of a face with $n$ dots showing up is proportional to $n$. What is the probability of the face showing 4 dots?
  • Let $X$ be a random variable with a probability distribution function $$f (x) = \begin{cases} 0.2 & \text{for  } |x|<1 \ 0.1 & \text{for } 1 < |x| < 4\ 0 & \text{otherwise} \end{cases}$$ The probability P (0.5 < x < 5) is ————-
  • Runs scored by batsmen in 5 one day matches are 50, 70, 82, 93, and 20. The standard deviation is ————-.
  • Find the median and mode of the messages received on 9 consecutive days 15, 11, 9, 5, 18, 4, 15, 13, 17.
  • $E (XY)=E (X)E (Y)$ if $x$ and $y$ are independent.
  • Mode is the value of $x$ where $f(x)$ is a maximum if $X$ is continuous.
  • A coin is tossed up 4 times. The probability that tails turn up in 3 cases is ————–.
  • If $E$ denotes the expectation the variance of a random variable $X$ is denoted as?
  • $X$ is a variate between 0 and 3. The value of $E(X^2)$ is ————-.
  • The random variables $X$ and $Y$ have variances of 0.2 and 0.5, respectively. Let $Z= 5X-2Y$. The variance of $Z$ is?
  • In a random experiment, observations of a random variable are classified as
  • A number of individuals arriving at the boarding counter at an airport is an example of
  • If $A$ and $B$ are independent, $P(A) = 0.45$ and $P(B) = 0.20$ then $P(A \cup B)$
  • If a fair dice is rolled twice, the probability of getting doublet is
  • If a fair coin is tossed 4 times, the probability of getting at least 2 heads is
  • If $P(B) \ne 0$ then $P(A|B) = $
  • The collection of all possible outcomes of an experiment is called
  • An event consisting of one sample point is called
  • An event consisting of more than one sample point is called
  • When the occurrence of an event does not affect the probability of occurrence of another event, it is called
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How to Perform Paired Samples t test in SPSS

In this post, we will learn about performing paired samples t test in SPSS. The paired samples t-test is a statistical hypothesis testing procedure used to determine whether the mean differences between two sets of observations are zero. In paired samples t-tests (also known as dependent samples) t-tests, each observation in one set is paired with the corresponding observation in another. In this test means/averages of two related groups are compared. By related we mean that the observations in the two groups are paired or matched in some way.

Points to Remember

The following are points that need to be remembered:

A Paired samples t-test can be used when two measurements are taken from the same individuals/objects/respondents or related units. The paired measurements can be:

  • Before and After Comparisons: A comparison of before and after situations, such as measuring blood pressure before and after taking medication.
  • Matched Pairs: Used when comparing the test scores of twins or blood relations.
  • Repeated Measures: When measuring a person’s happiness level at different points in time.

A paired samples t-test is also known as a dependent samples t-test, paired samples t-test, or repeated measures t-test.

Paired Samples t-test Cannot be used

Note that a paired samples t-test can only be used to compare the means for two related (paired) units having a continuous outcome that is normally distributed. This test is not appropriate when

  • The data is unpaired
  • There are more than two units/ groups
  • The continuous outcome is not normally distribution
  • The outcome is ordinal or ranked

Hypothesis for Paired Samples t test

The hypotheses for a paired/ dependent samples t-test can be stated as

$H_0:\mu_d = 0$ (the difference between the mean of pairs is zero (or equal) )
$H_1: \mu_d \ne$ (the difference between the mean of pairs is not zero (or different) )
$H_1: \mu_d < 0$ (upper tailed test)
$H_1: \mu_d > 0$ (lower-tailed test)

The test statistics for a paired samples t-test are as follows

$$t=\frac{\overline{d} }{\frac{s_d}{\sqrt{n}} }$$

where

  • $\overline{d}$ is the sample mean of the differences
  • $n$ is the sample size
  • $s_d$ is the sample standard deviation of the differences

Performing Paired Samples t test in SPSS

To run a paired samples t test in SPSS, click Analyze > Compare Means > Paired Samples t-test.

Paired Samples t-test in SPSS Analysis Procedure

Paired samples t-test dialog box, the user needs to specify the variables to be used in the analysis. The variables from the left side need to be moved from the paired variables box. A blue button in between both boxes may be used to shift the variables from left to right or right to left side. Note that the variables you specify in paired variables pan need to be in pair form.

Paired Samples t-test in SPSS Dilog box

In the above dialog box, the following are important points to follow:

  • Pair: The pair row (on the right side pane) represents the number of paired samples t-tests to run. More than one paired samples t-test may be run simultaneously by selecting multiple sets of matched variables.
  • Variables 1: The first variable represents the first match group (such as the before situation).
  • Variables 2: The second variable represents the second match group (such as the after situation).
  • Options: The options button can be used to specify the confidence interval percentage and how the analysis will deal with the missing values.
Paired Samples t-test in SPSS Options

Note that setting the confidence interval percentage does not have any impact on the calculation of the p-value.

Paired Samples t test Data Example

Consider the following example about 20 students’ academic performance by taking an examination before and after a particular teaching methodology.

Student NumberMarks before Teaching MethodologyMarks After Teaching Methodology
11822
22125
31617
42224
51916
62429
71720
82123
92319
101820

Testing the Assumptions of Paired Samples t-test

Before performing the Paired Samples t-test, it is better to test the assumptions (or requirements) of the paired samples t-test.

  • The dependent variable should be continuous (that is measured on interval or ratio level).
  • The dependent observations (related samples) should have the same subject/ objects, that is, the subjects in the first group are also in the second group.
  • Sampled data should be random from the respective population.
  • The differences between the paired values should follow the normal (or approximately) normal distribution
  • There should be no outliers in the differences between the two related groups.

Note that when testing the assumptions (such as normality, and outliers detection) related to paired samples t-test, one must use a variable that represents the differences between the paired values, not the original variables themselves.

Also note that when one or more assumptions for a paired samples t-test are not met, you may run the non-parametric test, Wilcoxon Signed Ranks Test.

Paired Samples t-test in SPSS for analysis

Output: Paired Samples T test

The SPSS will result in four tables:

  1. Paired Samples Statistics
    The paired samples statistics table gives univariate descriptive statistics (such as mean, sample, size, standard deviation, and standard error) for each variable entered as paired variables.
  2. Paired Samples Correlations
    The paired samples correlation table gives the bivariate Person correlation coefficient for each pair of variables entered.
  3. Paired Samples Test
    The paired samples test table gives the hypothesis test results with p-value and confidence interval of difference.
  4. Paired Samples Effect Sizes
    The paired sample Effect sizes tables give Cohen’s d and Hedges’ Correction values with confidence interval
Paired Samples t-test Output in SPSS

Interpreting the Paired Samples t test Output

From the “Paired Samples Test” the two-tailed p-value (0.121) is greater than 0.05 (level of significance), which means that the null hypothesis is accepted which means that there is no difference between marks before and after the teaching methodology. It means that improvement in marks is due to chance or random variation marely. The “Paired Samples Correlations” Table shows that the paired variables are correlated/ related to each other as the p-value for Pearson’s Correlation is less than 0.05.

The marks related to before and after teaching methodology are statistically and significantly related to each other, however, the average difference of marks between before and after teaching methodology is not statistically significant. The differences are due to change or random variation.

How to report the Paired Samples t-test Results

One might report the statistics in the following format: t(degrees of freedom) = t-value, p = significance level.
From the above example, this would be: t(9) = -1.714, p > 0.05. Due to the averages of the two situations and the direction of the t-value, one can conclude that there was a statistically non-significant improvement in marks due to the teaching methodology from 19.9 ± 2.685 marks to 21.50 ± 3.922 marks (p > 0.05). So, there is no improvement due to the teaching methodology.

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Properties of a Good Estimator

Introduction (Properties of a Good Estimator)

The post is about a comprehensive discussion of the Properties of a Good Estimator. In statistics, an estimator is a function of sample data used to estimate an unknown population parameter. A good estimator is both efficient and unbiased. An estimator is considered as a good estimator if it satisfies the following properties:

  • Unbiasedness
  • Consistency
  • Efficiency
  • Sufficiency
  • Invariance

Let us discuss these properties of a good estimator one by one.

Unbiasedness

An estimator is said to be an unbiased estimator if its expected value (that is mean of its sampling distribution) is equal to its true population parameter value. Let $\hat{\theta}$ be an unbiased estimator of its true population parameter $\theta$ then $\hat{\theta}$. If $E(\hat{\theta}) = E(\theta)$ the estimator ($\hat{\theta}$) will be unbiased. If $E(\hat{\theta})\ne \theta$, then $\hat{\theta}$ will be a biased estimator of $\theta$.

  • If $E(\hat{\theta}) > \theta$, then $\hat{\theta}$ will be positively biased.
  • If $E(\hat{\theta}) < \theta$, then $\hat{\theta}$ will be negatively biased.

Some examples of biased or unbiased estimators are:

  • $\overline{X}$ is an unbiased estimator of $\mu$, that is, $E(\overline{X}) = \mu$
  • $\widetilde{X}$ is also an unbiased estimator when the population is normally distributed, that is, $E(\widetilde{X}) =\mu$
  • Sample variance $S^2$ is biased estimator of $\sigma^2$, that is, $E(S^2)\ne \sigma^2$
  • $\hat{p} = \frac{x}{n}$ is an unbiased estimator of $E(\hat{p})=p$

It means that if the sampling process is repeated many times and calculations about the estimator for each sample are made, the average of these estimates would be very close to the true population parameter.

An unbiased estimator does not systematically overestimate or underestimate the true parameter.

Consistency

An estimator is said to be a consistent estimator if the statistic to be used as an estimator approaches the true population parameter value by increasing the sample size. OR
An estimator $\hat{\theta}$ is called a consistent estimator of $\theta$ if the probability that $\hat{\theta}$ becomes closer and closer to $\theta$, approaches unity with increasing the sample size.

Symbolically, $\hat{\theta}$ is a consistent estimator of the parameter $\theta$ if for any arbitrary small positive quantity $e$ or $\epsilon$.

\begin{align*}
\lim\limits_{n\rightarrow \infty} P\left[|\hat{\theta}-\theta|\le \varepsilon\right] &= 1\\
\lim\limits_{n\rightarrow \infty} P\left[|\hat{\theta}-\theta|> \varepsilon\right] &= 0
\end{align*}

A consistent estimator may or may not be unbiased. The sample mean $\overline{X}=\frac{\Sigma X_i}{n}$ and sample proportion $\hat{p} = \frac{x}{n}$ are unbiased estimators of $\mu$ and $p$, respectively and are also consistent.

It means that as one collects more and more data, the estimator becomes more and more accurate in approximating the true population value.

An efficient estimator is less likely to produce extreme values, making it more reliable.

Efficiency

An unbiased estimator is said to be efficient if the variance of its sampling distribution is smaller than that of the sampling distribution of any other unbiased estimator of the same parameter. Suppose there are two unbiased estimators $T_1$ and $T_2$ of the sample parameter $\theta$, then $T_1$ will be said to be a more efficient estimator compared to the $T_2$ if $Var(T_1) < Var(T_2)$. The relative efficiency of $T_1$ compared to $T_2$ is given by the ration

$$E = \frac{Var(T_2)}{Var(T_1)} > 1$$

Note that when two estimators are biased then MSE is used to compare.

A more efficient estimator has a smaller sampling error, meaning it is less likely to deviate significantly from the true population parameter.

An efficient estimator is less likely to produce extreme values, making it more reliable.

Sufficiency

An estimator is said to be sufficient if the statistic used as an estimator utilizes all the information contained in the sample. Any statistic that is not computed from all values in the sample is not a sufficient estimator. The sample mean $\overline{X}=\frac{\Sigma X}{n}$ and sample proportion $\hat{p} = \frac{x}{n}$ are sufficient estimators of the population mean $\mu$ and population proportion $p$, respectively but the median is not a sufficient estimator because it does not use all the information contained in the sample.

A sufficient estimator provides us with maximum information as it is close to a population which is why, it also measures variability.

A sufficient estimator captures all the useful information from the data without any loss.

A sufficient estimator captures all the useful information from the data.

Invariance (Property of Love)

If the function of the parameter changes, the estimator also changes with some functional applications. This property is known as invariance.

\begin{align}
E(X-\mu)^2 &= \sigma^2 \\
\text{or } \sqrt{E(X-\mu)^2} &= \sigma\\
\text{or } [E(X-\mu)^2]^2 &= (\sigma^2)^2
\end{align}

The property states that if $\hat{\theta}$ is the MLE of $\theta$ then $\tau(\hat{\theta})$ is the MLE of $\tau(\hat{\theta})$ for any function. The Taw ($\tau$) is the general form of any function. for example $\theta=\overline{X}$, $\theta^2=\overline{X}^2$, and $\sqrt{\theta}=\sqrt{\overline{X}}$.

Properties of a Good Estimator

From the above diagrammatic representations, one can visualize the properties of a good estimator as described below.

  • Unbiasedness: The estimator should be centered around the true value.
  • Efficiency: The estimator should have a smaller spread (variance) around the true value.
  • Consistency: As the sample size increases, the estimator should become more accurate.
  • Sufficiency: The estimator should capture all relevant information from the sample.

In summary, regarding the properties of a good estimator, a good estimator is unbiased, efficient, consistent, and ideally sufficient. It should also be robust to outliers and have a low MSE.

Properties of a good estimator

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