Question

A physchologist wishes to determine the variation in I.Q.s of the population in his city. He takes many random samples of size 64 . The standard error of the mean is found to be equal to \(2 .\) What is the population standard deviation?

Short answer

The population standard deviation can be found using the rearranged formula: Population standard deviation = Standard Error * squareroot of the Sample size. In this case, the population standard deviation is equal to 2 * squareroot of 64, which is 2 * 8 = 16. Therefore, the population standard deviation is 16.

Step-by-step solution

Understand the problem and the data provided

The psychiatrist wants to know the population standard deviation based on the standard error of the mean from the samples taken. The standard error of the mean is given as 2, and the sample size is given as 64.

Remember and rearrange the formula for standard error of the mean

The formula for standard error of the mean is: Standard Error = Population standard deviation / squareroot of the Sample size. To find the population standard deviation, we need to rearrange this formula to get: Population standard deviation = Standard Error * squareroot of the Sample size.

Insert the values and perform calculation

Now we can insert our values into the rearranged formula: Population standard deviation = 2 * squareroot of 64.

Calculate the square root of the sample size

Calculate the square root of the sample size, which is the square root of 64. This equals 8.

Compute final value for population standard deviation

Now, use the result in the transitioned formula to calculate the actual value for the Population standard deviation. Population standard deviation = 2 * 8 = 16 Hence, the population standard deviation in this case is 16.

Key concepts

Population Standard Deviation

Population standard deviation is a measure that tells us how much the individual data points in a population deviate from the mean (average) of the population. It's incredibly useful because it provides insight into the variability or spread of the population data.
Understanding this concept is crucial in fields that require accurate data interpretation, like psychology, where assessing IQ distribution can reveal much about societal trends.To calculate the population standard deviation when given the standard error of the mean, one must rearrange the formula for the standard error, which is:

• Standard Error = \(\frac{\sigma}{\sqrt{n}}\),

where \(\sigma\) is the population standard deviation, and \(n\) is the sample size. Rearranging gives:

• Population Standard Deviation = Standard Error * \(\sqrt{n}\).

In this way, the population standard deviation provides a foundational understanding of data distribution within a large group, helping to paint a detailed picture of variability.

Random Sampling

Random sampling is a technique used in statistics to ensure that every subset of a population has an equal chance of being selected. This is crucial in making sure that the sample represents the whole population well.
By using random sampling, biases in sample selection are minimized, leading to more accurate and reliable data analysis. In the context of the exercise, the psychologist takes many random samples of size 64 to ensure that the data truly reflects the population's intelligence quotient (IQ) distribution.
Here are some key advantages of random sampling:

• Minimizes bias, ensuring each member of the population has an equal chance of selection.
• Provides a basis for statistical inference and generalization of the results to the whole population.
• Allows researchers to calculate population parameters and test hypotheses effectively.

All these factors make random sampling a fundamental principle in conducting fair and accurate statistical analyses.

Sample Size

Sample size refers to the number of observations in a statistical sample. In statistical analyses, the sample size plays a significant role in the accuracy of the results and the general conclusions drawn about a population. A larger sample size generally provides more reliable data simply because it has more information.
In our exercise, each sample contains 64 individuals. This is a decent size, providing enough variability to estimate the population standard deviation accurately. The sample size affects:

• The precision of the standard error of the mean.
• The power of a statistical test—a larger sample size can detect smaller differences between groups.
• The confidence level of an interval estimate, where bigger samples typically lead to narrower confidence intervals.

Hence, understanding and choosing the correct sample size is paramount to achieving meaningful results in any research study.

Statistical Analysis

Statistical analysis is the science of collecting, exploring, and presenting large amounts of data to discover underlying patterns and trends. It's essential in making sense of data-heavy industries and academic fields.
In the scenario with IQ testing, statistical analysis helps draw conclusions about the population's intellectual capacity based on the sample data. Through statistical analysis, researchers can:

• Identify trends and relationships in data.
• Make generalizations about a population from sample data.
• Evaluate hypotheses and make predictions.

In this exercise, even though we're focused on finding the population standard deviation, this calculation is an integral part of broader statistical analyses that might involve comparison of IQ levels across different populations, age groups, or even over time.
In summary, statistical analysis allows data-driven decisions and insights, making it a cornerstone of modern research.