Sampling Theory In Statistics: Quiz! Test

The most commonly used confidence interval is 95 percent because it provides a balance between precision and reliability. A 95 percent confidence interval means that if we were to repeat the sampling process multiple times, we would expect the true population parameter to fall within the interval 95 percent of the time. This level of confidence is widely accepted in many fields and is considered a standard practice for statistical analysis.

As the sample size increases, the standard error decreases. This is because a larger sample size provides more data points, which leads to a more accurate estimate of the population parameter. With more data, the variability of the sample means decreases, resulting in a smaller standard error. In other words, as the sample size increases, the estimate becomes more precise and closer to the true population value, reducing the standard error.

A statistic is a function of sample observations because it is derived from analyzing a subset of data (the sample) rather than the entire population. Statistics are used to summarize and describe the data collected from the sample, providing insights and information about the population from which the sample was taken. By analyzing a sample, statisticians can make inferences and draw conclusions about the larger population.

A parameter is a characteristic that describes a population. It is a numerical value that summarizes the entire population, such as the mean or standard deviation. In contrast, a sample is a subset of the population, and statistics are used to describe the characteristics of a sample. Therefore, the correct answer is "Population" because parameters are specific to populations and not samples.

Simple random sampling is a probabilistic sampling method where each individual in the population has an equal chance of being selected for the sample. This means that every possible sample of a given size has an equal probability of being selected. This method ensures that the sample is representative of the population and reduces bias. Therefore, the correct answer is "A probabilistic sampling".

The sample mean is an MVUE (Minimum Variance Unbiased Estimator) for the population mean because it has the smallest variance among all unbiased estimators. It is also a consistent estimator because as the sample size increases, the sample mean converges to the population mean. Additionally, the sample mean is an efficient estimator because it achieves the Cramér-Rao lower bound, which means it has the smallest possible variance among all unbiased estimators. Finally, the sample mean is a sufficient estimator because it contains all the information in the sample needed to estimate the population mean. Therefore, the correct answer is "All of these".

Simple random sampling is very effective if the population is not very large because it allows for a manageable sample size that can still accurately represent the entire population. Additionally, it is effective if the population is not much heterogeneous because simple random sampling assumes that the population is homogeneous, meaning that each individual has an equal chance of being selected. Therefore, if the population is not much heterogeneous, simple random sampling can provide a representative sample.

Standard error is a measure of precision obtained by sampling. It represents the average amount of variability or fluctuation that can be expected in the sample mean compared to the population mean. It is calculated by dividing the standard deviation of the sample by the square root of the sample size. A smaller standard error indicates a more precise estimate of the population parameter.

The correct answer is a list of all possible samples that can be obtained by taking a random sample of size 2 with replacement from the population. In this case, the population contains the units 3, 6, and 1. With replacement means that after each selection, the unit is put back into the population, so it can be selected again. The answer includes all possible combinations of two units from the population, considering that the same unit can be selected multiple times.

An ideal estimator should be unbiased, meaning that on average it should give an estimate that is equal to the true value of the parameter being estimated. It should also be consistent, meaning that as the sample size increases, the estimate should converge to the true value. Efficiency refers to the estimator having a small variance, meaning that it is less likely to deviate far from the true value. Lastly, sufficiency means that the estimator should contain all the relevant information from the sample to make the estimate. Therefore, the correct answer is unbiasedness, consistency, efficiency, and sufficiency.

For an unknown parameter, there are many interval estimates that can be calculated. Interval estimates provide a range of values within which the true value of the parameter is likely to fall. These estimates can be calculated using different statistical methods and assumptions, resulting in a wide range of possible intervals. Therefore, there are many interval estimates that exist for an unknown parameter.

Purposive sampling is a non-probability sampling technique where the researcher deliberately selects specific individuals or cases that possess certain characteristics of interest. Unlike other sampling methods, the selection of participants in purposive sampling is based on the researcher's judgment and discretion. The researcher chooses participants who are believed to be the most informative or representative for the research objectives. Therefore, the decision of whom to include in the sample is entirely dependent on the sampler's judgment, making purposive sampling subject to the discretion of the sampler.

An infinite population refers to a population that is theoretically unlimited or has no definite boundary. In this case, the population of roses in Salt Lake City can be considered as an infinite population because there is no specific limit to the number of roses that can exist in the city. It is possible for the population to continue growing indefinitely, making it an example of an infinite population.

Standard error is a measure of the variability or dispersion of a statistic in repeated sampling. It quantifies the average amount of error or uncertainty in estimating a population parameter based on a sample. It is calculated as the standard deviation of the statistic. Therefore, the given answer "Standard deviation of a statistic" accurately describes the concept of standard error.

Stratified sampling is the correct answer because it allows for separate estimates of population means for different segments, ensuring that each segment is represented proportionately in the sample. This sampling method is particularly useful when there are distinct subgroups within the population that need to be accurately represented. Additionally, stratified sampling also provides an overall estimate by combining the estimates from each segment, giving a comprehensive picture of the population as a whole.

A sample survey is prone to both sampling errors and non-sampling errors. Sampling errors occur when the sample selected is not representative of the entire population, leading to biased results. Non-sampling errors, on the other hand, can occur due to various factors like data collection errors, respondent errors, or errors in data processing. These errors can affect the accuracy and reliability of the survey results, making it important to address and minimize them to ensure the validity of the findings.

Statistical decision about an unknown universe is taken on the basis of sample observations. This means that instead of collecting data from the entire population, a smaller subset, or sample, is taken and analyzed. By studying this sample, conclusions and decisions can be made about the larger population. This approach is often used because it is more practical and cost-effective than conducting a complete enumeration, which would require gathering data from every single individual or item in the population. A sampling frame is a list of all the potential individuals or items that could be included in the sample, and a sample survey is the process of collecting data from the chosen sample.

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If the sampling frame contains an undetected periodicity, the sampling method that would be affected the most is systematic sampling. Systematic sampling involves selecting every nth element from the sampling frame. If there is an undetected periodicity in the frame, it means that certain elements are repeated or occur at regular intervals. This can lead to a biased sample, as the systematic sampling method may consistently select elements that have the same periodicity, resulting in a non-random sample. In contrast, simple random sampling, stratified sampling, and multistage sampling are less likely to be affected by undetected periodicity in the sampling frame.

The standard error (SE) of the sample mean can be calculated using the formula SE = sqrt(variance/n), where variance is the population variance and n is the sample size. In this case, the population variance is known to be 25 and the sample size is 16. Therefore, the SE = sqrt(25/16) = sqrt(1.5625) = 1.25.

The sampling distribution refers to the probability distribution of a statistic. This means that it represents the range of possible values that a statistic can take on when calculated from different random samples drawn from the same population. It helps us understand the variability and likelihood of different sample statistics occurring. By studying the sampling distribution, we can make inferences about the population parameter that the statistic is estimating.

In Neyman's allocation, the sample size is determined by the proportion of the population size. This means that the larger the population size, the larger the sample size will be. This is because a larger population size requires a larger sample size in order to obtain a representative sample that accurately reflects the characteristics of the population. Therefore, the sample size is directly proportional to the population size in stratified sampling according to Neyman's allocation.

Sampling fluctuations refer to the natural variation in the values of a statistic. When we take a sample from a population, the values of the statistic calculated from different samples will vary due to random chance. This variation is known as sampling fluctuations. It is important to understand and consider sampling fluctuations when making inferences about the population based on the sample.

The correct answer implies that random sampling involves selecting units from a population in such a way that each unit has an equal chance of being chosen. This ensures that the sample is representative of the entire population and reduces bias. Random sampling is a fundamental principle in statistical research and allows for generalization of findings to the larger population.

The formula for variance of a random sample drawn without replacement is given by (m+1) (m-n)/12n. This formula takes into account the total number of elements in the set (m), the number of elements drawn (n), and calculates the variance using the formula for sampling without replacement. The other options do not correctly account for the sampling without replacement and/or do not use the correct formula for variance.

The number of all possible samples of size 2 that can be drawn from a population of 5 members with replacement can be calculated using the formula n^r, where n is the number of members in the population and r is the size of the sample. In this case, n = 5 and r = 2. Therefore, the number of possible samples is 5^2 = 25.

The possible samples from a population containing the units a, b, c, and d can be found by taking all possible combinations of two units. The answer choice (a, b), (a, c), (a, d), (b, c), (b, d), (c, d) includes all possible combinations of two units, ensuring that each unit is paired with every other unit exactly once. This accounts for all possible samples that can be taken without replacement from the population.

Based on the given information, 8 out of 100 life insurance policies were found to be insured for less than Rs.10,000. To estimate the number of policies in the whole lot that can be expected to be insured for less than Rs.10,000 at a 95% confidence level, we can use the formula for confidence interval for proportions.

Using this formula, we can calculate the lower and upper bounds of the confidence interval. The lower bound is calculated by subtracting the margin of error from the sample proportion, and the upper bound is calculated by adding the margin of error to the sample proportion.

In this case, the lower bound is 1058 and the upper bound is 2142. Therefore, we can expect that the number of policies in the whole lot insured for less than Rs.10,000 at a 95% confidence level will be between 1058 and 2142.

The correct answer is to infer about the unknown universe from a knowledge of a random sample drawn from it. This means that sampling allows us to make inferences about a larger population or universe based on information obtained from a randomly selected sample. By using a random sample, we can minimize bias and increase the likelihood that our findings will be representative of the entire population.

The given information states that the proportion of people in favor of liberalization lies between 0.683 and 0.817 with 95% confidence. This means that the sample proportion of 120 people falls within this range. To find the number of people in the group, we can use the formula for sample proportion: sample proportion = number of people in favor / total number of people. Rearranging the formula, we can calculate the total number of people by dividing the number of people in favor by the sample proportion. In this case, 120 / 0.817 = 146.8 and 120 / 0.683 = 175.6. Since the number of people cannot be in decimal, the closest option is 160. Therefore, the number of people in the group is 160.

The 95% confidence limits for the average height of all the students forming the population can be calculated using the formula:

Confidence interval = sample mean ± (critical value * standard error)

In this case, the sample mean is 1.75m and the standard error can be calculated by dividing the standard deviation (0.35m) by the square root of the sample size (100).

The critical value for a 95% confidence level with a sample size of 100 can be found in a standard normal distribution table and is approximately 1.96.

Plugging in the values, we get:

Confidence interval = 1.75m ± (1.96 * (0.35m / √100))

Simplifying the equation, we get:

Confidence interval = 1.75m ± 0.0686m

Therefore, the 95% confidence limits for the average height of all the students forming the population is [1.68m, 1.82m].

The Law of Statistical Regularity states that a large sample drawn at random from a population would possess the characteristics of the population on an average. This means that when a large enough sample is taken, the sample will represent the population accurately in terms of its characteristics. This principle is based on the assumption that random sampling allows for a representative sample to be obtained, which in turn allows for generalizations to be made about the entire population.

The given information states that the sample size is 17 and the mean is 52. It also provides the sum of squares of deviation from the mean, which is 160. Based on this information, we can calculate the standard deviation of the sample, which is the square root of the sum of squares of deviation divided by the sample size minus 1. Using this standard deviation, we can calculate the margin of error for a 99% confidence interval, which is the critical value multiplied by the standard deviation divided by the square root of the sample size. Finally, we can calculate the confidence limits by subtracting and adding the margin of error to the mean. The resulting confidence limits are [42.77, 61.23].

The lower value of the amount that the company will have to pay in insurance during the year can be calculated by multiplying the number of policies (1500) by the average amount of each policy (Rs.2000). This gives a total of Rs.3,000,000. Since 99,000 out of 100,000 individuals survive at age 31, it can be assumed that only 1% of the policies will result in a payout. Therefore, the company will have to pay out a minimum of 1% of Rs.3,000,000, which is Rs.30,000. However, since the question asks for the lower value, the correct answer is Rs.6000.

The estimate of the standard error of the sample mean can be calculated using the formula: standard deviation / square root of the sample size. In this case, the standard deviation (SD) is given as 4 and the sample size is 10. Plugging these values into the formula, we get 4 / √10 = 0.63. Therefore, the estimate of the standard error of the sample mean is 0.63.

The sample standard deviation is a biased estimator for the population standard deviation because it tends to underestimate the true value of the population standard deviation. This bias occurs because the sample standard deviation uses the sample mean in its calculation, which is an estimate of the population mean. Since the sample mean is not always equal to the population mean, this introduces a bias in the estimation of the standard deviation.

The upper confidence limit can be calculated using the formula:
Upper confidence limit = mean + (z * (SD / sqrt(n)))
where mean is the sample mean, z is the z-score corresponding to the desired confidence level, SD is the sample standard deviation, and n is the sample size.

In this case, the sample mean is Rs.7500, the z-score corresponding to a 90% confidence level is 1.645 (approximated as 2.58 for simplicity), the sample standard deviation is Rs.80, and the sample size is 82.

Plugging these values into the formula:
Upper confidence limit = 7500 + (2.58 * (80 / sqrt(82)))
Upper confidence limit ≈ 7500 + (2.58 * 8.82)
Upper confidence limit ≈ 7500 + 22.7416
Upper confidence limit ≈ 7522.7416

Rounded to two decimal places, the upper confidence limit is Rs.7520.98, which matches the given answer.

Systematic sampling adds flexibility to the sampling process because it allows for a systematic and organized approach to selecting samples. With systematic sampling, the researcher selects every nth element from the population, which provides a structured method for sampling. This method allows for a more efficient and convenient sampling process, especially when the population is large and spread out. It also ensures that every element in the population has an equal chance of being selected, which helps to reduce bias and increase the representativeness of the sample.

The standard error of the sample mean is a measure of how much the sample mean is likely to vary from the population mean. It is calculated by dividing the population standard deviation by the square root of the sample size. In this case, the population standard deviation is known to be 5 and the sample size is 25. Therefore, the standard error of the sample mean is 5 / sqrt(25) = 5 / 5 = 1. Hence, the correct answer is 1.

When a sample is taken from a population with replacement, it means that each member of the population has an equal chance of being selected for the sample each time. In this case, the sample observations are 1, 3, and 5. To estimate the standard error of the sample mean, we need to calculate the standard deviation of the population divided by the square root of the sample size. Since the population size is 10 and the sample size is 3, the standard error of the sample mean is estimated to be 2.00.

The 95% confidence interval for the population mean is calculated using the formula: Mean ± (1.96 * Standard Error). In this case, the lower confidence limit (LCL) is 48.04 and the upper confidence limit (UCL) is 51.96. Therefore, the mean is (48.04 + 51.96) / 2 = 50. The standard error can be calculated as (UCL - LCL) / (2 * 1.96), which gives us (51.96 - 48.04) / (2 * 1.96) = 1.96. The population variance is equal to the square of the standard error, so the population variance is 1.96^2 = 3.8416. However, since the sample size is 100, we need to divide the population variance by the sample size, resulting in 3.8416 / 100 = 0.038416. Therefore, the value of the population variance is approximately 0.038416, which is closest to 10.

The estimate of the standard error (SE) of the proportion of rotten oranges in the sample is 0.01. This means that the proportion of rotten oranges in the population is estimated to be within 0.01 of the proportion observed in the sample. The SE is a measure of the variability or uncertainty in the estimate, and a smaller SE indicates a more precise estimate. In this case, the SE of 0.01 suggests that the estimate of the proportion of rotten oranges is relatively precise.