How To Calculate The Sample Size Given The Confidence Level & Margin of Error

The Organic Chemistry Tutor
19 Jun 202006:42
EducationalLearning
32 Likes 10 Comments

TLDRThis video explains how to calculate sample sizes for statistical experiments. The first example shows the formula for margin of error and manipulates it to derive a formula for sample size. Given values for mean, standard deviation, margin of error, and confidence level, the number of students is calculated. The second example calculates maximum sample size to determine the proportion of high schoolers attending college after graduation. Using the formula for sample size with a proportion, 0.5 is chosen for the sample proportion p-hat to maximize the sample size. With the provided confidence level and margin of error, the maximum sample size is calculated to be 385 students.

Takeaways
  • ๐Ÿ˜€ The video discusses calculating sample size for statistical experiments
  • ๐Ÿ“Š Key formulas covered: margin of error, sample size based on mean/standard deviation, sample size for proportion
  • ๐Ÿงฎ Worked through two sample problems step-by-step: one for sample size based on mean and standard deviation, and one for sample size based on proportion
  • ๐Ÿ”ข For 95% confidence level, z-score is 1.96
  • ๐Ÿ“ˆ To maximize sample size for proportion, use 0.5 for p-hat in sample size formula
  • ๐Ÿค“ Provided references for reviewing statistical concepts like calculating z-scores
  • ๐Ÿ‘ฉโ€๐Ÿซ Explained how to determine best p-hat value to use in sample size for proportion formula
  • ๐Ÿ“ Emphasized calculating maximum sample size and rounding up
  • โœ๏ธ Sample size calculations allow determining required number of subjects for statistical experiments
  • ๐Ÿ“š Offered additional statistics video recommendations in description for reference
Q & A
  • What is the formula for calculating margin of error?

    -The margin of error is equal to the z-score times the standard deviation divided by the square root of the sample size.

  • What is the z-score for a 95% confidence level?

    -The z-score for a 95% confidence level is 1.96.

  • How do you calculate sample size given the mean, standard deviation, and margin of error?

    -The formula is: Sample size = (z-score)^2 * (standard deviation)^2 / (margin of error)^2.

  • What formula is used to calculate sample size for a proportion?

    -The formula is: Sample size = (z-score)^2 * (sample proportion) * (1 - sample proportion) / (margin of error)^2.

  • Why should the maximum sample proportion be used in the sample size calculation?

    -Using the maximum sample proportion (0.5) will give the largest possible sample size needed to achieve the desired margin of error.

  • What is the best sample proportion to use when the true proportion is unknown?

    -When the true proportion is unknown, using 0.5 as the sample proportion will maximize the sample size calculation.

  • What are the steps for calculating the needed sample size for a proportion?

    -1. Identify the confidence level and get the z-score. 2. Determine the acceptable margin of error. 3. Use 0.5 as the sample proportion. 4. Plug into the sample size formula and calculate.

  • Why do we need to round the sample size up to a whole number?

    -The sample size represents the number of individuals in the sample. Since you can't sample a fractional number of people, the calculated sample size needs to be rounded up to the nearest whole number.

  • What factors determine how large a sample needs to be?

    -The key factors are the desired confidence level, the acceptable margin of error, the estimated variability in the population, and for proportions - the estimated proportion value itself.

  • When calculating sample size, why is a larger sample better?

    -A larger sample size produces a smaller margin of error, which leads to more reliable and precise estimates of the true population parameter.

Outlines
00:00
๐Ÿ“Š Calculating Sample Size with Known Mean and Standard Deviation

This segment introduces the method to calculate the sample size needed for a statistical experiment given a specific set of parameters: a mean score of 84, a standard deviation of 15, and a margin of error of 5.367 at a 95% confidence level. The video explains the formula for margin of error as the product of the z-score and the standard deviation divided by the square root of the sample size. It then rearranges the formula to solve for the sample size, using a z-score of 1.96 for the 95% confidence level. By substituting the known values into the formula, it calculates the sample size to be 30, indicating the number of students in the class for the statistical experiment.

05:03
๐Ÿ“ˆ Determining Maximum Sample Size for Proportion Estimates

The second problem focuses on finding the maximum sample size for estimating the proportion of high school students planning to enroll in college after graduation, given a 95% confidence level and a 5% margin of error. The video explores how to calculate sample size for proportion-based studies, introducing a different formula involving the square of the z-score, the estimated proportion (p-hat), and the margin of error. It emphasizes the choice of 0.5 for p-hat to maximize the product of p-hat and (1 - p-hat), leading to the largest possible sample size. Following the formula, it calculates a sample size of 384.16, which is rounded up to 385, to achieve the desired margin of error with a 95% confidence level.

Mindmap
Keywords
๐Ÿ’กSample Size
Sample size refers to the number of observations or individuals included in a statistical sample. In the context of the video, calculating the sample size is crucial for ensuring statistical experiments are accurately representative of the whole population. For instance, the video demonstrates how to calculate the sample size needed to achieve a specific margin of error and confidence level in two scenarios: one involving the average test score of students and another concerning the proportion of high school students planning to enroll in college.
๐Ÿ’กStandard Deviation
Standard deviation is a measure of the amount of variation or dispersion of a set of values. The video uses this concept to calculate the sample size for a class's average test scores, where a standard deviation of 15 indicates how much individual test scores deviate from the average score of 84. This statistic helps in understanding the spread of data points and is pivotal in determining the margin of error.
๐Ÿ’กMargin of Error
Margin of error represents the range of values below and above the sample statistic in a confidence interval. The video explains how a margin of error of 5.367 and 5% is used to calculate sample sizes for different scenarios, emphasizing its role in indicating the expected range within which the true population parameter lies with a certain level of confidence.
๐Ÿ’กConfidence Level
A confidence level is a percentage that indicates how often the true population parameter would fall within the calculated confidence interval if the same experiment were repeated multiple times. The video discusses a 95% confidence level, signifying that the researcher is 95% confident the true mean falls within the margin of error of the observed statistics.
๐Ÿ’กZ-Score
A Z-score, or standard score, is a numerical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. The video uses a Z-score of 1.96 for calculations related to a 95% confidence level, demonstrating its use in determining the sample size needed for the statistical experiments.
๐Ÿ’กProportion
Proportion refers to the fraction of the total that possesses a certain attribute. In the video, Sally is interested in the proportion of high school students planning to enroll in college. This concept is essential for calculating sample sizes in studies where the outcome is categorical, such as 'yes' or 'no' answers.
๐Ÿ’กP-hat
P-hat represents the sample proportion, an estimate of the true proportion in the entire population. The video illustrates how to use P-hat in the formula for calculating the maximum sample size, particularly emphasizing the value of 0.5 for P-hat to maximize the sample size for a given margin of error and confidence level.
๐Ÿ’กFormula
Formulas are mathematical expressions used to solve problems. The video introduces specific formulas for calculating sample size based on the margin of error, standard deviation, Z-score, and P-hat. These formulas provide a systematic approach for determining the number of observations needed for statistical reliability.
๐Ÿ’กStatistical Experiment
A statistical experiment involves the collection and analysis of data to observe and measure outcomes. The video discusses conducting statistical experiments to determine the average test scores of students and the proportion of high school students planning to go to college, showcasing the practical application of statistical theories in real-life scenarios.
๐Ÿ’กRounding
Rounding is the process of adjusting numbers to a specified degree of precision. The video concludes with rounding the calculated sample size to the nearest whole number, as the sample size must be an integer. This step is essential in statistical calculations to ensure practical applicability of the results.
Highlights

Proposed a new deep learning model called BERT that achieved state-of-the-art performance on multiple NLP tasks

BERT uses bidirectional training of Transformers, allowing the model to learn contextual relations between words

The pretrained BERT model can be fine-tuned for downstream tasks using additional output layers

BERT large model achieves F1 score of 93.2 on SQuAD question answering dataset, outperforming human performance

BERT advances state-of-the-art on GLUE natural language understanding benchmark by a large margin

Proposes masked language modeling objective to enable bidirectional representation learning in BERT

BERT representations capture rich contextual information about word usage and meaning

BERT has wide applicability for transfer learning on many NLP tasks with minimal task-specific architecture modifications

BERT has become a foundational model for pretraining in NLP, enabling significant progress on benchmark tasks

Code and pretrained models released publicly, enabling wide adoption of BERT in industry and academia

BERT highlights the effectiveness of large-scale bidirectional pretraining for learning universal language representations

Authors foresee BERT driving progress in conversational AI and language understanding

BERT has motivated follow-up work investigating bidirectional pretraining techniques and model architectures

The pretrained BERT model is highly generalizable and applicable to many languages beyond English

BERT is considered a milestone in NLP, opening new research directions in pretrained language models

Transcripts
Rate This

5.0 / 5 (0 votes)

Thanks for rating: