6) Explain the term “Statistical Inference”. How is the statistical inference based upon the estimation of parameters.

Statistical inference is the process of drawing conclusions about a population based on information obtained from a sample. It involves using sample statistics (e.g., mean, standard deviation) to estimate population parameters (the corresponding values in the population). The accuracy of these estimations is crucial for making valid inferences.

7) Indicate the role of standard error for statistical generalisation.

The standard error plays a crucial role in statistical generalization by providing a measure of the variability or precision of sample statistics. A smaller standard error indicates a more precise estimate of the population parameter, increasing confidence in the generalizability of findings from the sample to the population.

8) Differentiate between significance of statistics and confidence interval of fiduciary limits.

• Significance of Statistics: Focuses on determining whether an observed effect (e.g., difference between sample means) is likely to have occurred due to chance alone. It involves calculating a test statistic and comparing it to a critical value based on a chosen significance level (e.g., .05 or .01).

• Confidence Interval (Fiduciary Limits): Provides a range of values within which the population parameter is estimated to lie with a certain degree of confidence (e.g., 95% or 99%). It considers the variability of the sample statistic (standard error) to express the uncertainty in the estimation.

9) Enumerate the various uses of Standard Error of the statistics.

• Assessing the reliability of sample statistics: A smaller standard error indicates greater reliability and precision of the estimate.

• Estimating population parameters: Standard error is used to construct confidence intervals around sample statistics, providing a range within which the population parameter is likely to fall.

• Determining sample size: Standard error helps in calculating the necessary sample size to achieve a desired level of precision in the estimation of population parameters.

• Testing the significance of differences between groups: The standard error of the difference between means is used to determine whether observed differences are statistically significant or likely due to chance.

10) What type of errors can occur while interpreting the results based on test of significance? How we can overcome these errors?

• Type I Error (α error): Rejecting the null hypothesis when it is actually true (false positive).

• Type II Error (β error): Accepting the null hypothesis when it is actually false (false negative).

To mitigate these errors:

• Choose an appropriate significance level (α) based on the study's context and the consequences of each type of error.
• Ensure adequate sample size to increase the power of the test and reduce the risk of a Type II error.
• Carefully consider the study design, data quality, and potential confounding factors that might influence the results.

11) A Sample of 100 students with mean score 26.40 and SD 5.20 selected randomly from a population. Determine the .95 and .99 for confidence intervals for population true mean.

To determine the .95 and .99 confidence intervals for the population true mean, we'll use the formula: Confidence Interval = M ± (Z × SEM)

Where:

• SEM = σ / √N
• Z = the z-score corresponding to the desired confidence level (.95 or .99)

Calculating SEM: SEM = 5.20 / √100 = 0.52

For .95 Confidence Interval: Z = 1.96 (from the z-table)

Confidence Interval = 26.40 ± (1.96 × 0.52) = 25.38 to 27.42

For .99 Confidence Interval:

Z = 2.58 (from the z-table) Confidence Interval = 26.40 ± (2.58 × 0.52) = 25.06 to 27.74

12) A small sample of 10 cases with mean score 175-50 and ∑ =5.82 selected randomly. Compute finding limits of parameter mean at .05 and .01 level of confidence.

Since this is a small sample (N < 30), we'll use the "t" distribution instead of the "Z" distribution. The formula for the confidence interval is similar:

Confidence Interval = M ± (t × SEM)

Where:

• SEM = s / √N − 1
• s = the sample standard deviation
• t = the t-score corresponding to the desired confidence level (.95 or .99) and degrees of freedom (df = N − 1)

Calculating s:

The sources do not explicitly state how to calculate the sample standard deviation (s) from the given information. However, if "∑" represents the sum of squared deviations from the mean, then s can be calculated as:

s = √(∑ / N − 1) = √(5.82 / 9) = 0.807

Calculating SEM:

SEM = 0.807 / √10 − 1 = 0.269

For .95 Confidence Interval:

df = 10 − 1 = 9

t = 2.262 (from the t-table for df = 9 and .95 confidence level)

Confidence Interval = 175.50 ± (2.262 × 0.269) = 174.90 to 176.10

For .99 Confidence Interval:

t = 3.250 (from the t-table for df = 9 and .99 confidence level)

Confidence Interval = 175.50 ± (3.250 × 0.269) = 174.62 to 176.38

13) Explain Estimate.

A value calculated from a sample that is used to approximate a population parameter. It's essentially an educated guess about the population characteristic based on the available sample data. For instance, the sample mean (M) is an estimate of the population mean (µ).

14) Explain Parameter.

A numerical value that describes a characteristic of a population. It's the true value we aim to estimate using statistical inference. Examples include the population mean (µ), population standard deviation (σ), and population correlation (η).

15) Explain Statistics.

Numerical values that are calculated from sample data. They are used to summarize and describe the sample, and they serve as estimates for population parameters. Common statistics include the sample mean (M), sample standard deviation (s), and sample correlation (r).

16) Explain Sampling Error.

The difference between a sample statistic and the corresponding population parameter. It arises because a sample is only a subset of the population, and therefore may not perfectly represent the entire population. A smaller sampling error suggests a more representative sample.

17) Explain Measurement Error.

Error introduced during the process of measuring a variable. It can be due to limitations of the measuring instrument, inconsistencies in data collection, or human error in recording observations.

18) Explain Standard Error.

A measure of the variability or precision of a sample statistic. It quantifies how much sample statistics are expected to fluctuate from one sample to another, drawn from the same population. A smaller standard error indicates a more precise estimate of the population parameter.

19) What is the general formula to know the standard errors of the various statistical measures?

Standard Error (SE) = Standard Deviation / √N

Where:

• Standard Deviation refers to the standard deviation of the sampling distribution of the statistic.
• N represents the sample size.

20) What do you mean by significance and levels of significance?

Significance in statistics refers to the likelihood that an observed effect or relationship in the data is not due to chance alone. A statistically significant result indicates that the observed effect is unlikely to have occurred randomly, suggesting a genuine effect in the population.

Level of significance (α): It is a pre-determined threshold used in hypothesis testing to determine the probability of rejecting the null hypothesis when it is actually true (Type I Error). Common levels of significance in behavioral sciences are 0.05 (5%) and 0.01 (1%).

21) In behavioural sciences, which levels of confidence are considered

In behavioral sciences, the most frequently used levels of confidence are 95% and 99%. These correspond to significance levels of 0.05 and 0.01, respectively.

22) What is the difference between significance of statistics and confidence interval for true statistics?

• Significance of Statistics: Focuses on determining whether an observed effect is statistically significant, meaning it's unlikely to have arisen due to random chance. It involves hypothesis testing and calculating p-values.

• Confidence Interval for True Statistics: Provides a range of values within which the true population parameter is estimated to lie with a certain level of confidence. It quantifies the uncertainty associated with the estimation of the parameter.

23) The mean achievement score of a random sample of 400 psychology students is 57 and D.D. is 15? Determine how dependable is the sample mean?

The sample mean of 400 psychology students is fairly dependable. With a large sample size (N = 400) and a standard deviation of 15, the standard error of the mean would be relatively small, indicating a high level of precision in the estimate of the population mean.

24) A sample of 80 subjects has the mean = 21.40 and standard deviation 4.90. Determine how far the sample mean is trustworthy to its Mpop.

With a sample of 80 subjects, a mean of 21.40, and a standard deviation of 4.90, the sample mean is reasonably trustworthy to its population mean. While not as large as the previous example, a sample size of 80 is still considered relatively large, suggesting a moderate level of reliability in estimating the population mean.

25) What is the concept of Degree of Freedom (df)?

Degrees of Freedom (df): In essence, degrees of freedom represent the number of independent pieces of information available to estimate a statistical parameter. They account for the constraints imposed on the data when calculating statistics. 

26) Why we consider the (df) which determining the reliability or trustworthiness of the statistics.

Degrees of freedom are considered when determining the reliability or trustworthiness of statistics because they affect the shape of sampling distributions and the critical values used in hypothesis testing. The appropriate degrees of freedom need to be used to ensure accurate statistical inferences. 

27) What is the difference to calculate the standard error of Mean of Large Size and Small Size samples.

The key difference in calculating the standard error of the mean for large and small samples lies in the use of degrees of freedom.

• Large samples (N > 30): The standard error of the mean is calculated as: SEM = s / √N, where s is the sample standard deviation. Degrees of freedom are not explicitly considered in the formula because their impact is negligible with large samples.

• Small samples (N ≤ 30): The standard error of the mean is calculated as: SEM = s / √(N-1). Here, degrees of freedom (N - 1) are explicitly used in the denominator to account for the greater uncertainty associated with small samples.

28) Given M =26.40, ó = 5.20 and N=100 compute:

The fiduciary limits of True Mean at 99% confidence interval

The fiduciary limits of Population Mean at .95 confidence interval.

• Fiduciary limits of the true mean at a 99% confidence interval: 25.06 to 27.74
• Fiduciary limits of the population mean at a 95% confidence interval: 25.36 to 27.44

29) The mean of 16 independent observations of a certain magnitude is 100 and S.D is 24. At .05 confidence level what are the fiduciary limits of the True Mean.

With a mean of 100, SD = 24, and N = 16, the fiduciary limits of the true mean at a 95% confidence level are 87.68 to 112.32.

30) Suppose it is known that S.D of the scores in a certain population is 20. How many cases would we in a sample in order that the S.E of the sample mean be σ 2.

If the population standard deviation (σ) is 20 and the desired standard error of the mean (SEM) is 2, the required sample size is 100. This ensures that the sample mean is a reasonably precise estimate of the population mean.

31) What do you mean by significance of the difference in two means?

The significance of the difference in two means refers to whether the observed difference between the means of two samples is likely due to chance or represents a real difference between the populations from which the samples were drawn.

32) Define Standard Error of the difference of the two sample means.

The standard error of the difference of the two sample means (SEDM) is a measure of how much variability is expected in the difference between two sample means if they were repeatedly drawn from the same population. It quantifies the precision of the estimate of the difference between the population means.

33) Define Sampling distribution of the differences of Means of two Samples.

The sampling distribution of the differences of means of two samples is a theoretical distribution of the differences between means that would be obtained if you repeatedly drew pairs of samples from the same population(s) and calculated the difference between their means.

34) What should be the mean value of sampling distribution of the difference of the means?

If the two populations have the same mean, the mean value of the sampling distribution of the difference of the means should be zero.

35) What indicates S.E.DM?

SEDM, or σDM, indicates the variability or spread of the differences between sample means. A smaller SEDM suggests a more precise estimate of the difference between population means.

36) What do you mean by Ho, Define it.

The null hypothesis (H0) is a statement of no effect or no difference. In the context of comparing two means, it states that there is no true difference between the means of the two populations. The purpose of hypothesis testing is to determine whether there is enough evidence from the sample data to reject the null hypothesis.

37) What are the assumptions on which testing of the difference of two Mean is based?

• The variable or trait being measured is normally distributed in the population(s).
• The samples are drawn randomly from the population(s).

38) What do you mean by One Tail Test and Two Tail Test? When these two tails are used?

• One-tailed test: Used when the alternative hypothesis specifies the direction of the expected difference (e.g., Group A will have a higher mean than Group B).
• Two-tailed test: Used when the alternative hypothesis does not specify the direction of the difference, only that the means are different.

The choice depends on the research question and the directionality of the hypothesis being tested.

39) What is meant by uncorrelated and correlated sample means?

• Uncorrelated (Independent) sample means: Computed from different, unrelated groups of participants.
• Correlated (Dependent) sample means: Computed from the same group of participants measured at different times or from matched pairs of participants.

40) How you can define “Critical Ratio” and “t” Ratio?

Both the critical ratio (CR) and t-ratio are statistics that measure how far the observed difference between two sample means is from the expected difference under the null hypothesis, expressed in standard error units. They help assess the statistical significance of the difference.

41) What is the difference between “CR” and “t”?

• CR: Typically used with large samples (N > 30) where the sampling distribution of the mean difference is assumed to be approximately normal.

• t: Used with small samples (N ≤ 30) where the sampling distribution follows a t-distribution, which accounts for the greater uncertainty associated with smaller samples.

42) What is the difference between C.R. and Z Score?

CR and z-score are essentially the same when dealing with large samples (N > 30). They both represent the standardized difference between a sample mean and a population mean, or between two sample means, under the assumption of a normal distribution.

43) How can you define the standard Error of the Difference of Means?

The standard error of the difference of means is a measure of the variability expected in the difference between two sample means if they were repeatedly drawn from the same population(s). It indicates the precision of the estimate of the difference between the population means.

44) What formula you will use in the following conditions:

• Two large independent samples: CR = (M1 – M2) / σDM, where σDM = √(σ21 / N1 + σ22 / N2).

• Two large correlated samples: t = (M1 – M2) / √(σ2M1 + σ2M2 – 2r12σM1σM2).

• Two small independent samples: t = (M1 – M2) / SEDM, where SEDM = SD√(1/N1 + 1/N2).

• Two small correlated samples: t = (M1 – M2) / (SDD /√N), where SDD is the standard deviation of the differences between paired scores.

45) What do you mean by independent samples?

Independent samples are drawn from separate, unrelated groups of participants. There is no pairing or matching between participants in the two groups.

46) What do you mean by correlated samples?

• Correlated samples involve paired or matched observations from the same group of participants or from two groups that have been deliberately matched on certain characteristics.
• This creates a dependency between the observations in the two samples.