- Please define each of the following terms: sampled population, random sampling, convenient sampling, judgmental sampling, stratified random sampling, consistency in sampling, relative efficiency. Explain why a sample is of probabilistic nature
A sampled population is a specific small group of people from where the data is collected. The sample must be random to apply statistics in studying things about the population. However, the size of the sample is less than the size of the population taken. One example is asking 30 randomly chosen students in a university class what their main course is. The sample is 30, while the population is all the learners in the class (Lin, L. 2018).
Random sampling: is part of the variety selecting sampling technique in which every sample has an equal probability of being chosen. A sample that is chosen randomly is meant as unbiased and a representation of the total population.
Convenient sampling in statistics, it’s referred to as availability sampling. It is a general type of non-probability sampling method that majorly depends on collecting data from members of a population ready to study.
Relative efficiency refers to the comparative performance and a measure of the quality of an estimator. It depicts how much the set of alternatives /entities are efficient in employing based on the inputs and output. However, experiments, a more efficient, or test need a keen observation than a less efficient way to achieve a given result.
Explain why a sample is of probabilistic nature
The sample is probabilistic because it involves itself as a reflective of probability. For example, if a targeted sample is pinpointed, then the confidence interval will be of no use. This explains why, by nature, a sample is just probabilistic
.
- What is it meant by the term “parameter of a population”? Explain why a population can
be represented by a random variable.
It’s a description of a fixed measure of the whole group of people (population). It is a summary of the number that illustrates the entire group.
Explain why a random variable can represent a population.
The reason for this, random variable valued is unknown and has a precise value/continuous. The function assigns values to every experiment outcome (Shaliz & Rinaldo, 2013).
3.What is a point estimate and an unbiased point estimate? Explain how the sample means can be an unbiased estimate of the population mean. How do you justify that the sample variance is an unbiased estimate of the population variance? What is the sampling requirement in the latter case? Provide a numerical example of estimating the mean, the variance, and the standard deviation.
The point estimate involves using data samples to calculate a single value that serves as a “greater guess” of the population’s unknown parameter.
While unbiased point estimates, the population’s mean is the difference between the expected values, this is when the estimated value and the value of the estimated parameters are equal, and then the estimator is considered unbiased.
Explain how the sample means can be an unbiased estimate of the population mean.
The term unbiased is the act of being unfair to others. In statistics, when the population’s parameters are underestimated or overestimated, then the statistics are termed as unbiased. One can take different steps to ensure the statistic is accurate to reflect and unbiased depending on the population you are studying. However, one should take samples according to the sound statistical practices by avoiding the measured error by ensuring that the data is collected in an unbiased way. And lastly by avoiding any unrepresentative samples by ensuring you exclude specific population individual
4.Please define each of the following terms, discuss the applicability and significance of each: sample statistic, standard error, sampling distribution, and central limit theorem. Include hypothetical examples for better clarity.
Sample statistics. It is a piece of the subset that contains the characteristics of a larger population. It’s manageable and a smaller version of a larger group. It’s applicable in statistics testing when the population size is large for the test, including all possible individuals or observations. The significance in it the result is unlikely to change; however, it tells us what the probability is the relationship we think is due only to random chance.
Standards error. Is a measure of the accuracy in statistics with which a sample distribution represents a population by the use of standard deviation
A sampling distribution is the probability distribution of a statistic gotten5 from various samples drawn from a specific population. Its significance is useful in making inferences about the overall population.
The central limit theorem does explain that the sampling distribution of the mean approaches a normal distribution as the same size increases. Its significance is that it tells people that sampling distribution approaches normality as the sample size increases. The average of sample means will itself be the population mean. As the standard deviation of any given sample means equal, the standard error of the population means.
- What is the z statistic, and what qualifies a statistic to be z statistic based on the central limit theorem and the basic properties of normal distributions? What are the limitations of the central limit theorem, and how some of these limitations are bypassed? For example, the z statistic is the sampling distribution in estimating a proportion.
z-score(z) is the test statistics. The equation that defines it is z=(p−P)σ, p is the hypothesized value in the population proportion in the null hypothesis, the initial p is the sample proportion, and σ is the standard deviation of the distribution in sampling. The central limit theorem states that whether one has the population with mean and standard deviation and takes sufficiently large random samples from the population with replacement, then the sample’s distribution will be approximately normal. is the sampling
- What distribution in estimating the variance of a population? What are the properties of this distribution?
A sample distribution is the variance of the mean’s sampling distribution: the population variance divided by N. is about the distribution of statistics obtained from a large figure of samples drawn from a specific population. The sampling distribution of any specific population is all about distributing frequencies of a range of different outcomes for a population statistic (Hossain,2020).
Properties.
The distribution shape is symmetric and approximately normal.
The center of distribution is very close to the real population mean.
There are no outliers or other significant deviation from the overall pattern.
What is the alternative of the z statistic for customarily distributed populations which
Eliminates other limitations of the central limit theorem?
7 What is the z statistic’s alternative for customarily distributed populations, which eliminates some limitations of the central limit theorem? How is this sampling distribution constructed as a combination of a z distribution and a chi-squared distribution? What are the properties of this distribution?
. The t- distribution can be formed by taking samples of the same size from an average population. The relative frequency distribution of these t statistics is the t distribution. It turns out that the t statistics can be computed a few different ways on samples draw in several different situations and still have the same relative frequency distribution. However, the t distribution becomes useful in making different inferences.so t distribution is the most crucial link samples and populations used by statisticians.
References
Lin, L. (2018). Bias caused by sampling error in meta-analysis with small sample sizes. PloS one, 13(9), e0204056.
Hossain, S. (2020). Logic, Probability Theory, and their Application to Legal Reasoning.
Shalizi, C. R., & Rinaldo, A. (2013). Consistency under-sampling of exponential random graph models. Annals of Statistics, 41(2), 508.