MCQs Probability Distributions Quiz 5

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This Quiz contains the MCQs Probability Distributions Quiz. It covers events, experiments, mutually exclusive events, collectively exhaustive events, sure events, impossible events, addition and multiplication laws of probability, concepts related to discrete and continuous random variables, probability distribution and probability density functions, characteristics and properties of probability distributions, discrete probability distribution, and continuous probability distributions, etc.

Online MCQs about Probability Distributions with Answers

1. The normal distribution is also classified as

 
 
 
 

2. If $N$ is the population size, $n$ is the sample size, $p$ is the probability of success, $K$ is the number of successes stated in the population, and $k$ is the number of observed successes, then the parameters of the binomial distribution are

 
 
 
 

3. An oil company conducts a geological study that indicates that an exploratory oil well should have a 0.25 probability of striking oil. The company is interested in finding the probability that the 3rd strike comes on the 6th well drilled. Which distribution will be used?

 
 
 
 

4. A random variable $X$ has a binomial distribution with $n=9$, the variance of $X$ is

 
 
 
 

5. In binomial probability distribution, the formula for calculating standard deviation is

 
 
 
 

6. When can we use a normal distribution to approximate a binomial distribution?

 
 
 
 

7. An oil company conducts a geological study that indicates that an exploratory oil well should have a 20% chance of striking oil. The company is interested in finding the probability that the first strike comes on the third well drilled. Which distribution distribution will be used?

 
 
 
 

8. If $X$ follows Geometric distribution with parameter $p$ (probability of success) then the Mean of $X$ is

 
 
 
 

9. In normal distribution, the proportion of observations that lies between 1 standard deviation of the mean is closest to

 
 
 
 

10. The parameters of hypergeometric distributions are Note that $N$ is the population size, $n$ is the sample size, $p$ is the probability of successes, $K$ is a number of successes stated in the population, $k$ is the number of observed successes.

 
 
 
 

11. For Beta distribution of 2nd kind, the range of $X$ is

 
 
 
 

12. The formula of the mean of uniform or rectangular distribution is as

 
 
 
 

13. The mean deviation of a normal distribution is

 
 
 
 

14. Which of the distributions has a larger variance than its mean

 
 
 
 

15. The distribution of the square of the standard normal random variable will be

 
 
 
 

16. For beta distribution of 1st kind, the range of $X$ is

 
 
 
 

17. Themean of the Poisson distribution is 9 then its standard deviation is

 
 
 
 

18. The Chi-Square distribution is a special case of

 
 
 
 

19. In binomial probability distributions, the dependents of standard deviations must include

 
 
 
 

20. In any normal distribution, the proportion of observations that are outside $\pm$ standard deviation of the mean is closest to

 
 
 
 

Probability distributions are the foundation for various statistical tests like hypothesis testing. By comparing observed data to a theoretical distribution (the null hypothesis), we can assess the likelihood that the data arose by chance.

Probability distributions are crucial tools in data analysis. They help identify patterns, outliers, and relationships between variables. Furthermore, many statistical models depend on specific probability distributions to function accurately.

Probability Distributions

Online MCQs Probability Distributions Quiz

  • In binomial probability distributions, the dependents of standard deviations must include
  • In binomial probability distribution, the formula for calculating standard deviation is
  • The formula of the mean of uniform or rectangular distribution is as
  • The normal distribution is also classified as
  • The mean deviation of a normal distribution is
  • The Chi-Square distribution is a special case of
  • Which of the distributions has a larger variance than its mean
  • For Beta distribution of 2nd kind, the range of $X$ is
  • The mean of the Poisson distribution is 9 then its standard deviation is
  • In normal distribution, the proportion of observations that lies between 1 standard deviation of the mean is closest to
  • For beta distribution of 1st kind, the range of $X$ is
  • The parameters of hypergeometric distributions are Note that $N$ is the population size, $n$ is the sample size, $p$ is the probability of successes, $K$ is a number of successes stated in the population, $k$ is the number of observed successes.
  • If $N$ is the population size, $n$ is the sample size, $p$ is the probability of success, $K$ is the number of successes stated in the population, and $k$ is the number of observed successes, then the parameters of the binomial distribution are
  • An oil company conducts a geological study that indicates that an exploratory oil well should have a 0.25 probability of striking oil. The company is interested in finding the probability that the 3rd strike comes on the 6th well drilled. Which distribution will be used?
  • If $X$ follows Geometric distribution with parameter $p$ (probability of success) then the Mean of $X$ is
  • The distribution of the square of the standard normal random variable will be
  • A random variable $X$ has a binomial distribution with $n=9$, the variance of $X$ is
  • In any normal distribution, the proportion of observations that are outside $\pm$ standard deviation of the mean is closest to
  • When can we use a normal distribution to approximate a binomial distribution?
  • An oil company conducts a geological study that indicates that an exploratory oil well should have a 20% chance of striking oil. The company is interested in finding the probability that the first strike comes on the third well drilled. Which distribution distribution will be used?
Probability distributions Quiz

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How to Split Data File in SPSS?

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In SPSS (Statistical Packages for Social Sciences) split file option lets the user to splits the data into separate groups for analysis based on the values of one or more grouping variables. If user select multiple grouping variables, the cases are grouped by each variable within categories of the preceding variable on the groups based on list. Let us learn about the step-by-step procedure to Split Data file in SPSS.

How to Split Data File in SPSS

Suppose you want to take the separate mean of male and female (groups/ categories from gender variable) then one may use split file option.

  • First Open the data file you want to split.
  • Second, from the menu bar, click the Data Menu and then Split File Option (Data -> Split File)
Split Data File in SPSS Menu

The following dialog box “Split File” will appears. Click on the radio button title “Organize output by Groups” after clicking the Grouping variable from left pan.

Split File in SPSS Dialog Box Options
  • Select the Gender Varaible (or the grouping variable you want to split) in the dialog box at the left pan and clikc on the arrow at the “Groups based on” box.
Split File in SPSS
  • Click the OK button. Now, subsequent analyses will reflect the split.
  • The data in data windows will be logical splitted. One can run requierd descriptive and inferential analsysi of the splitted data.

Split File Off

  • The most important point is to get back to ‘normal’ where the data are not split, go back to Data/Split Files… and select the option ‘Analyze All cases.’
  • Press OK. It will show SPLIT FILE OFF. Then you can get back output of data without splitting the files.

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Leverage Influential Point and Outlier: Diagnostics (2024)

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In this post, a discussion about diagnostics for a Leverage Influential point and outlier will be made. In a regression analysis, certain observations may play a role in influencing the outcomes of the fitted model and its estimates. These observations may be classified as outliers, leverage, and influential points.

Outlier Leverage Influential Point

The explanation of outlier leverage influential point is described as under:

  • Outliers: An outlier is an extreme observation that differs considerably from the other observations. An outlier may be due to the recording error and the model cannot explain them. However, outlier(s) may contain some important information. An outlier may be in $x$-space, $y$-space, or both.
  • Leverage: An unusual $x$ value is called a leverage point. The leverage point affects the model summary statistics (such as $R^2$, standard error, etc.), but has little impact on the estimates of the regression coefficients. A leverage point has an unusual predictor value and is different from the bulk of the observations.
  • Influence: An unusual $y$ value (and may be an extreme $x$ value), is called an influence point. An influence point has a noticeable impact on the estimated regression coefficients and may change the direction of the slope.
Diagnostics for Outliers leverage and influential points
image taken from: https://www.cbsd.org/

Diagnostics for Outlier Leverage Influential Point

There are some methods to detect/ identify the outlier leverage influential point

Outliers

Outliers must be treated very carefully. Outliers may be detected by examining the

  • Normal Quantile Plots (departer from normality)
  • Residual Plots (magnitude of the residuals)
  • Scaled residuals (a potential outlier if magnitudes > 3)
Outlier Detection using Box Plot

Leverage Point

The diagonal elements of the “hat matrix” have an important role in detecting influential observations. $$h_{ii} = x’_i (X’X)^{-1}x_i,$$ where $X$ is matrix of regressors and $x’_i$ is the ith row of the $X$ matrix.

A large diagonal element is an indicator of influential observation as they are remote in $x$-space. Any observation exceeding the average size of the diagonal element of the hat matrix ($\overline{h} = \frac{p}{n}=2h$) is considered as a leverage point, where $p$ is the number of parameters in the model.
It is also useful to observe the studentized residuals in conjunction with $h_{ii}$ (that is, look for large hat diagonal and large residual values).

Note that not all of the leverage points are influential unless they have large residuals. Therefore, observations having large $h_{ii}$ values and large residuals are likely to be R.

Influential Points

  • Cook’s Distance: The Cook’s Distance is the Deletion Diagnostic that is used to measure the influence of the $i$th observation by removing it from the regression analysis. It is based on all $n$ points, $\hat{\beta}, and the estimates based on the deletion of the $i$th point, $\hat{\beta}_{(i)}$.
  • DFBETAS is another Deletion Diagnostic used to measure how the change in each of the $\hat{\beta}j$ is due to influential observation. A large value of DFBETAS indicates that the $i$th observation is considerably an influential observation on the $j$th regression coefficient. If $|DFBETAS{j, i} > \frac{2}{\sqrt{n}}$ then the $i$th observation warrants further examination.
  • DFFITS is another deletion diagnostic measure used to measure the deletion influence of the $i$th observation on the predicted or fitted values. DFFITS is the number of standard deviations that the fitted values change if ith observations are removed. If $|DFFITS_i|>\frac{2}{\sqrt{\frac{p}{n}}}$ then the $i$th observation warrants further examination.

Note that the case deletion diagnostics do not provide any information about the overall prediction of the estimation. However, the performance of the model can be measured by using the Generalized Variance (GV) and Covariance Ratio.

In summary, the Outliers, Leverage Points, and Influential Observations are certain data points (observations) that deviate (distant) from the expected patterns. On the other hand, the outliers are extreme values that lie far away from the other data points, while leverage points exert a strong influence on the regression models.

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