Problem Description:
The statistical analysis assignment focuses on various statistical concepts, including probability distributions, hypothesis testing, confidence intervals, and effect sizes. Here's a breakdown of the solutions provided:
Solutions:
1. Normal Random Variable Properties
Problem:
Explain the notation, PDF, and properties of the normal random variable.
Solution:
Describes the Probability Density Function (PDF) and properties of the normal random variable.
2. Normal Random Variable Questions
Problem:
Solve questions related to the normal random variable.
Solution:
Addresses a practical scenario involving a normal distribution, calculates probabilities, and uses the transformation equation.
3. Normal Distribution Functions
Problem:
Explain differences between dnorm(), pnorm(), qnorm(), and rnorm() functions.
Solution:
Describes functions related to normal distribution, including density, cumulative density, quantile, and random sampling.
4. More Normal Random Variable Questions
Problem:
Use pnorm(), qnorm(), and rnorm() to solve questions related to the normal random variable.
Solution:
Demonstrates the use of functions for probability calculations and random sampling.
5. Sampling Concepts
Problem:
Explain what is a simple random sample.
Solution:
Defines a simple random sample as a subset with equal probability of selection.
6. Sampling with and without replacement
Problem:
Explain the difference between sampling with replacement and without replacement.
Solution:
Differentiate between the two sampling methods.
7. Sampling Techniques
Problem:
Explain the differences between stratified sampling, snowball sampling, and convenience sampling.
Solution:
Describe various sampling techniques and their applications.
8. The Law of Large Numbers
Problem:
Explain what the law of large numbers is.
Solution:
The law of large numbers, in probability and statistics, states that as a sample size grows, its mean gets closer to the average of the whole population.
9. Central Limit Theorem
Problem:
Explain what the central limit theorem is.
Solution:
The central limit theorem says that the sampling distribution of the mean will always be normally distributed, as long as the sample size is large enough.
10. Difference Between the Law of Large Numbers and the Central Limit Theorem
Problem:
Explain the difference between the law of large numbers and the central limit theorem.
Solution:
The Law of Large Numbers states that when the sample size tends to infinity, the sample mean equals the population mean. In contrast, the Central Limit Theorem tells us that as the sample size tends to infinity, the distribution of the sample means approaches the normal distribution.
11. Sampling Distribution of the Sample Mean
Problem:
Explain what is the sampling distribution of the sample mean.
Solution:
A "sampling distribution of sample means" is the distribution of potential probabilities for the arithmetic average of a sample of a population.
12. Central Limit Theorem and Natural Traits
Problem:
Explain how the central limit theorem is related to the binomial distribution and the distribution of some naturally occurring traits (e.g., height and intelligence).
Solution:
A biologist may measure the height of 30 randomly selected plants and then use the sample mean height to estimate the population mean height. If the biologist finds that the sample mean height of the 30 plants is 10.3 inches, then her best guess for the population mean height will also be 10.3 inches.
13. Problems Related to the Central Limit Theorem
Problem:
Solve problems related to the central limit theorem.
Solution:
In a study, it was reported that the mean of mobile users is 30 years and the standard deviation is 12. Taking a sample size of 100, what is the mean and standard deviation for the sample mean ages of tablet users?
Solution: Since the sample mean will tend to the population mean, thus, the mean is 30. The sample standard deviation is σ/root(n) = 12 / 10 = 1.2
14. Estimation vs. Hypothesis Testing
Problem:
Explain the difference between estimation and hypothesis testing.
Solution:
A hypothesis test is used to determine whether or not a treatment has an effect, while estimation is used to determine how much effect.
15. Point Estimation vs. Interval Estimation
Problem:
Explain the difference between point estimation and interval estimation.
Solution:
The process of using a single statistic as an estimator of the population parameter is known as point estimation, whereas interval estimation involves the determination of an interval based on two numbers that are expected to contain the true value of the population parameter in its interior with a given probability or confidence.
16. Compute Estimations of Population Mean and Variance
Problem:
Compute the estimation of population mean and variance.
Solution:
xˉ±z∗nσ
17. Properties of a Good Point Estimation
Problem:
Explain the properties of a good point estimation.
Solution:
- Unbiasedness: An estimator is said to be unbiased if its expected value is identical to the population parameter being estimated.
- Consistency: If an estimator approaches the parameter closer as the sample size increases, it is considered consistent.
- Efficiency: Efficiency refers to the sampling variability of an estimator. If two unbiased estimators exist, the one with the smaller variance is considered more efficient.
- Sufficiency: An estimator is said to be sufficient if it conveys as much information as possible about the parameter contained in the sample.
18. Properties of Estimations of Population Mean and Variance
Problem:
Explain the properties of the estimations of population mean and population variance.
Solution:
The sample mean, ˉxˉ, tells you the average measurement from your sample; the standard deviation (SD) indicates how much variation there is in the data around the sample mean; the standard error (SE) represents the uncertainty associated with viewing the sample mean as an estimate of the mean of the whole population, μ.
19. Confidence Interval vs. Credible Interval
Problem:
Explain the difference between confidence interval and credible interval.
Solution:
Credible intervals capture our current uncertainty in the location of the parameter values and can be interpreted as a probabilistic statement about the parameter. In contrast, confidence intervals capture the uncertainty about the interval we have obtained.
20. Derivation of Confidence Interval
Problem:
Explain the derivation of the confidence interval.
Solution:
A confidence interval is the mean of your estimate plus and minus the variation in that estimate.
21. Computing Confidence Intervals with Different Percentages
Problem:
Be able to compute 80%, 90%, 95%, and 99% confidence intervals for the mean, assuming the population variance is known.
Solution:
xˉ±z∗nσ
For example, with a 95 percent confidence interval, you have a 5 percent chance of being wrong.
22. Purpose of Confidence Intervals
Problem:
Explain the purpose of confidence intervals.
Solution:
The purpose of confidence intervals is to provide a range of values for our estimated population parameter rather than a single value or a point estimate.
23. Interpretation of Confidence Interval
Problem:
Interpret the confidence interval.
Solution:
A 95% confidence interval of the mean [9,11][9,11] suggests you can be 95% confident that the population mean is between 9 and 11.
24. Computing the Confidence Interval for the Mean
Problem:
Compute the confidence interval for the mean.
Solution:
CI=ˉ±∗CI=xˉ±z∗nσ
25. Statistical Hypothesis
Problem:
Explain what is a statistical hypothesis.
Solution:
A statistical hypothesis is a formal claim about a state of nature structured within the framework of a statistical model.
26. Null Hypothesis vs. Alternative Hypothesis
Problem:
Explain the difference between the null hypothesis and the alternative hypothesis.
Solution:
A null hypothesis is what the researcher tries to disprove, whereas an alternative hypothesis is what the researcher wants to prove.
27. One-Tailed vs. Two-Tailed Hypothesis Testing
Problem:
Explain the difference between one-tailed (a.k.a., directional) vs. two-tailed (a.k.a., non-directional) hypothesis testing.
Solution:
A one-tailed test is used to ascertain if there is any relationship between variables in a single direction, i.e., left or right. In contrast, the two-tailed test is used to identify whether or not there is any relationship between variables in either direction.
28. Test Statistic
Problem:
Explain what a test statistic is.
Solution:
A test statistic is a number calculated by a statistical test. It describes how far your observed data is from the null hypothesis of no relationship between variables or no difference among sample groups.
29. Sampling Distribution of a Test Statistic
Problem:
Explain what the sampling distribution of a test statistic means.
Solution:
The sampling distribution of a statistic is a type of probability distribution created by drawing many random samples of a given size from the same population.
30. Sampling Distribution of the Sample Proportion
Problem:
Explain what the sampling distribution of the sample proportion is.
Solution:
The Sampling Distribution of the Population Proportion gives you information about the population proportion.
31. Type I Error and Type II Error
Problem:
Explain what Type I errors and Type II errors are.
Solution:
A Type I error (false-positive) occurs if an investigator rejects a null hypothesis that is true in the population; a Type II error (false-negative) occurs if the investigator fails to reject a null hypothesis that is false in the population.
32. Power in Hypothesis Testing
Problem:
Explain what power means in hypothesis testing.
Solution:
Power is the probability of making a correct decision (to reject the null hypothesis) when the null hypothesis is false.
33. α and β in Hypothesis Testing
Problem:
Explain what α and β denote in hypothesis testing.
Solution:
Alpha (α) is the probability of a Type I Error, and Beta (β) is the probability of a Type II Error.
34. Logic Behind the Sign Test
Problem:
Explain the logic behind the sign test.
Solution:
The sign test involves denoting values above the median of a continuous population with a plus sign and the ones falling below the median with a minus sign to test the hypothesis that there is no difference in medians.
35. Conducting a Sign Test
Problem:
Conduct a sign test using dbinom(), pbinom(), or binom.test() functions.
Solution:
RCopy code
binom.test(9, 24, 1/6)
36. Sign Test vs. Binomial Test
Problem:
Explain why the sign test is also known as the binomial test.
Solution:
They both use the theory that the two outcomes have equal probabilities.
37. Interpretation of p-value
Problem:
Interpret p-value.
Solution:
P-values are often interpreted as the risk of rejecting the null hypothesis of the test when the null hypothesis is true.
38. Purpose of the p-value
Problem:
Explain the purpose of the p-value.
Solution:
The p-value is simply a way to use surprise as a basis for making a reasonable decision.
39. P-Hacking
Problem:
Explain what p-hacking is.
Solution:
P-hacking is the act of misusing data analysis to show that patterns in data are statistically significant when, in reality, they are not.
40. Publication Bias
Problem:
Explain what publication bias is.
Solution:
Publication bias is defined as the failure to publish the results of a study based on the direction or strength of the study findings.
41. Motivation for P-Hacking
Problem:
Explain why researchers are motivated to p-hack.
Solution:
Researchers are not rewarded for being right, but rather for publishing a lot.
42. Purpose of One-Sample Z-Test and T-Test
Problem:
Explain the purpose of the one-sample z-test and t-test.
Solution:
A Z-test is used when we know the standard deviation of the comparison population (σ); a t-test is used when we do not have that information and must estimate the standard deviation from the sample (S).
43. Hypotheses Construction for One-Sample Z-Test
Problem:
Construct the hypotheses of a one-sample z-test.
Solution:
- Select the appropriate statistic (one-tailed or two-tailed).
- Determine the null hypothesis and alternative hypothesis.
- Determine the level of significance.
- Find the critical value.
- Calculate the test statistics.
44. General Procedure in One-Sample Z-Test
Problem:
Explain the general procedure in a one-sample z-test.
Solution:
- Select the appropriate statistic (one-tailed or two-tailed).
- Determine the null hypothesis and alternative hypothesis.
- Determine the level of significance.
- Find the critical value.
- Calculate the test statistics.
45. Critical Region and Critical Value
Problem:
Explain what the critical region and critical value are.
Solution:
A critical value is a number that separates two regions: the critical region, which is the set of values of the test statistic that leads to a rejection of the null hypothesis, and the acceptance region, which is the set of values for which the null hypothesis is not rejected.
46. CLT in Z-Test
Problem:
Explain how the CLT is used in the z-test.
Solution:
The z-test is best used for sample sizes greater than 30 because, under the Central Limit Theorem (CLT), as the number of samples increases, the samples are considered to be approximately normally distributed.
47. Hypotheses of One-Sample T-Test
Problem:
Construct the hypotheses of a one-sample t-test.
Solution:
The null hypothesis for a one-sample t-test can be stated as: "The population mean equals the specified mean value.
48. General Procedure in One-Sample T-Test
Problem:
Explain the general procedure in a one-sample t-test.
Solution:
- Calculate the test statistics.
- Decide on the level of significance.
- Find the value from the t-distribution based on the level of significance.
- Compare the value of the statistic with the t value.
49. Assumptions of Z-Test and T-Test
Problem:
Explain the assumptions of the one-sample z-test and t-test, and explain the differences between their assumptions.
Solution:
The difference between the z-test and the t-test is in the assumption of the standard deviation σ of the underlying normal distribution. A z-test assumes that σ is known; a t-test does not.
50. CLT in Z-Test and T-Test
Problem:
Explain how CLT is used in the z-test and t-test.
Solution:
The z-statistic is used to test for the null hypothesis concerning differences between population means or proportions when the population standard deviation is known, data belong to a normal distribution, and the sample size is large (greater than 30). T-tests are used when the population standard deviation is unknown, the data belong to a normal distribution, and the sample size is small (less than 30).
51. Construction of (t) Random Variable from Normal Random Variable
Problem:
Explain how the (t) random variable is constructed from the normal random variable.
Solution:
The t-distribution can be formed by taking many samples (all possible samples) of the same size from a normal population. For each sample, the same statistic called the t-statistic, is calculated.
52. Differences Between t-distribution and z-distribution
Problem:
Explain the differences between t-distribution and z-distribution.
Solution:
The standard normal or z-distribution assumes that the population standard deviation is known. The t-distribution is based on the sample standard deviation.
53. Using pt, qt, and rt Functions
Problem:
Use pt, qt, and rt functions.
Solution:
RCopy code
pt(-0.785, 14), qt(0.99, df = 20), rt(n = 5, df = 20)
54. Degrees of Freedom in (t) Distribution
Problem:
Explain what affects degrees of freedom (df) in (t) distribution and how df affects the shape of the t distribution.
Solution:
The degrees of freedom affect the shape of the graph in the t-distribution; as the df gets larger, the area in the tails of the distribution gets smaller. As df approaches infinity, the t-distribution will look like a normal distribution.
55. Paired-Samples vs. Independent-Samples
Problem:
Explain the difference between paired samples and independent-samples.
Solution:
Paired-samples t-tests compare scores on two different variables but for the same group of cases; independent-samples t-tests compare scores on the same variable but for two different groups of cases.
56. Similarity Between One-Sample and Paired-Sample Tests
Problem:
Explain the similarity between one-sample and paired-sample tests.
Solution:
They are performed to test a significant difference in the mean of the two groups.
57. Procedure for Paired-Samples T-Test
Problem:
Explain the general procedure for conducting a paired-sample t-test.
Solution:
- Calculate the sample mean.
- Calculate the sample standard deviation.
- Calculate the test statistic.
- Calculate the probability of observing the test statistic under the null hypothesis. This value is obtained by comparing t to a t-distribution with (n − 1) degrees of freedom.
58. Conducting Paired-Sample T-Test
Problem:
Conduct a paired-sample t-test using the t-test function.
Solution:
t.test(formula = score ~ time, alternative = "greater", mu = 0, paired = TRUE, var.equal = TRUE, conf.level = 0.95)
59. Difference Between Paired-Samples and Independent Samples
Problem:
Explain the difference between paired samples and independent samples.
Solution:
Paired-sample t-tests compare scores on two different variables but for the same group of cases; independent-sample t-tests compare scores on the same variable but for two different groups of cases.
60. Test-Statistic and Its Sampling Distribution for Independent Samples T-Test
Problem:
Explain how the test statistic and its sampling distribution are derived for the independent samples t-test.
Solution:
Calculate the sample mean. Calculate the sample standard deviation. Calculate the test statistic. Calculate the probability of observing the test statistic under the null hypothesis. This value is obtained by comparing t to a t-distribution with (n − 1) degrees of freedom.
61. Independent-Samples T-Test by Hand and Using R
Problem:
Conduct an independent-sample t-test by hand and using R.
Solution:
Hand:
- 5 observations in each sample: n1=n2=5
- The mean of sample 1: ¯x1=0.02
- The mean of sample 2: ¯x2=0.06
- Variances of both populations: σ21=σ22=1
- Test statistics: (0.02−0.06−0)/0.632 = −0.063
- Critical value: ±zα/2=±z0.025=±1.96
62. Confidence Interval for Population Mean with Unknown Variance
Problem:
Compute the confidence interval for the population mean when the population variance is unknown.
Solution:
The t.test function in R also gives the estimate of the confidence interval for the population mean when the population variance is unknown.
63. Confidence Intervals in t.test and binom.test
Problem:
Explain what the confidence intervals printed out in the t-test and binom. test are estimating.
Solution:
The binomial confidence interval is a measure of uncertainty for a proportion in a statistical population, while the t-test confidence interval refers to the probability that a population parameter will fall between two set values.
64. Setting Different Confidence Interval Percentages
Problem:
Set different confidence interval percentages in t.test and binom. test.
Solution:
RCopy code
prop.test(x=56, n=100, conf.level=.95, correct=FALSE) t.test(x, conf.level = 0.80)
65. Relationship Between Confidence Interval and Hypothesis Testing
Problem:
Explain the relationship between confidence interval and hypothesis testing.
Solution:
Confidence intervals and hypothesis tests are similar in that they are both inferential methods that rely on an approximated sampling distribution.
66. Decision Based on Confidence Interval in a Non-Directional Test
Problem:
Use the confidence interval to decide whether to reject the null hypothesis in a non-directional test.
Solution:
If we have μ0=5 and our confidence interval is (5.01, 5.25), and since 5 lies outside the range of the confidence interval, we decide to reject H0.
67. Factors Affecting the Power of a Test
Problem:
Explain factors that affect the power of a test.
Solution:
The four primary factors that affect the power of a statistical test are the significance level, the difference between group means, the variability among subjects, and the sample size.
68. Practical Significance vs. Statistical Significance
Problem:
Explain the difference between practical significance and statistical significance.
Solution:
While statistical significance shows that an effect exists in a study, practical significance shows that the effect is large enough to be meaningful in the real world.
69. Meaning of Effect Size
Problem:
Explain what effect size means.
Solution:
An effect size is a value measuring the strength of the relationship between two variables in a population.
70. Unstandardized vs. Standardized Effect Sizes
Problem:
Explain the difference between unstandardized and standardized effect sizes.
Solution:
Standardized effect size statistics remove the units of the variables in the effect, while the units remain in the original units of the variables for the unstandardized effect size.
71. Computing Unstandardized Effect Sizes
Problem:
Compute the unstandardized effect sizes.
Solution:
The mean temperature in condition 1 was 2.3 degrees higher than in condition 2. The standardized effect size statistic would divide that mean difference by the standard deviation.
72. Computing Cohen’s d for Different T-Tests
Problem:
Compute Cohen’s d for one-sample, paired-sample, and independent samples t-tests.
Solution:
Provide the specific values or data needed for computation.
73. Interpreting Cohen’s d
Problem:
Interpret Cohen’s d.
Solution:
An effect size of 0.5 means the value of the average person in Group 1 is 0.5 standard deviations above the average person in Group 2.
74. Construction of Chi-Squared (χ2) Random Variable
Problem:
Explain how the chi-squared (χ2) random variable is constructed from standard normal random variables.
Solution:
The chi-squared distribution is obtained as the sum of the squares of k-independent, zero-mean, unit-variance Gaussian random variables.
75. Using pchisq, qchisq, and rchisq Functions
Problem:
Use pchisq, qchisq, and rchisq functions.
Solution:
RCopy code
pchisq(0.86404, df=2), qchisq(p=0.95, df=13), rchisq(n=1000, df=5)
76. Chi-Squared Goodness-of-Fit Test
Problem:
Conduct the chi-squared goodness-of-fit test.
Solution:
RCopy code
chisq.test(x=observed, p=expected)