In an essay of 250-500 words, thoroughly address the following items and respond to the related questions:

1.Define the term standard deviation. Why is it important to know the standard deviation for a given sample? What do researchers learn about a normal distribution from knowledge of the standard deviation? A sample of n=20 has a mean of M = 40. If the standard deviation is s=5, would a score of X= 55 be considered an extreme value? Why or why not?
2.Hypothesis testing allows researchers to use sample data, taken from a larger population, to draw inferences (i.e., conclusions) about the population from which the sample came. Hypothesis testing is one of the most commonly used inferential procedures. Define and thoroughly explain the terms null hypothesis and alternative hypothesis. How are they used in hypothesis testing?
3. Define the term standard error. Why is the standard error important in research using sample distributions? Consider the following scenario: A random sample obtained from a population has a mean of µ=100 and a standard deviation of σ = 20. The error between the sample mean and the population mean for a sample of n = 16 is 5 points and the error between a sample men and population mean for a sample of n = 100 is 2 points. Explain the difference in the standard error for the two samples.

It is computed by the average distance from the average or average of all the averages for multiple sets of data. It is important because it measures the variability in the set of data. It helps in finding how close the values of data set are to the mean. We compute the z-score to find whether it is an extreme value. The Z-score = (55 – 40)/5 = 3. Since the z-score lies outside the two standard deviations from the mean, therefore it is considered to be an extreme value. Standard deviation is a measure of dispersion. It is computed by the average distance from the average or average of all the averages for multiple sets of data. We compute the z-score to find whether it is an extreme value. The Z-score = (55 – 40)/5 = 3. Since the z-score lies outside the two standard deviations from the mean, therefore it is considered to be an extreme value. The traditional hypothesis testing requires setting up two competing statements known as null hypothesis and alternative hypothesis. They are mutually exclusive and exhaustive. Null hypothesis Ho: The finding occurred by chance. It represents theory either because it is believed to be true or because it is used as a basis of argument, but not proved. Alternative Hypothesis H1: The finding did not occur by chance. It is a statement of what a hypothesis test is set to be established. They are used in hypothesis testing when the question is interested in simplified into two competing claims between there are two choices, the null hypothesis (Ho) against the alternative hypothesis (Ha). These two competing claims are not treated on equal basis; special consideration is given to the null hypothesis. Standard error is the standard deviation divided by the square root of sample size. It is known as the standard deviation of sampling distribution. It measures how much sample statistic varies from sample to sample. It is important because how much sampling fluctuate with the statistics. The inferential statistics involves confidence and significance testing which are based on the standard errors. The standard error depends on the sample size. Larger the sample size, smaller the standard error. The difference between the two samples is as the sample size increases, the standard error decreases because standard error and sample size are inversely related.

To answer the first question:

1. Standard Deviation: Standard deviation is a measure of the dispersion or spread of a set of values or data points. It tells us how much the values in a sample deviate from the mean (average) of the sample. A higher standard deviation indicates greater variability, while a lower standard deviation suggests less variability.

The importance of knowing the standard deviation for a given sample lies in the insights it provides about the data. It allows us to understand how closely the values in the sample cluster around the mean. A smaller standard deviation indicates that the data points are relatively close to the mean, while a larger standard deviation implies more widely scattered data.

Researchers learn about a normal distribution from knowledge of the standard deviation by observing the characteristics of the Bell curve shape. In a normal distribution, approximately 68% of the data falls within one standard deviation from the mean, about 95% falls within two standard deviations, and roughly 99.7% falls within three standard deviations. This allows for a better understanding of the proportion of data falling within a specific range of values.

For the sample with a mean of 40 and a standard deviation of 5, we can determine if a score of 55 is considered an extreme value. First, we calculate the z-score of 55. The formula for calculating the z-score is:

z = (X - M) / s

where X is the value, M is the mean, and s is the standard deviation. Plugging in the values:

z = (55 - 40) / 5 = 3

A z-score of 3 represents a value that is three standard deviations above the mean. Since a value that is three standard deviations away from the mean contains approximately 99.7% of the data in a normal distribution, a score of 55 would not be considered an extreme value in this case. It falls within a range that is still within the normal distribution.

Moving on to the second question:

2. Null hypothesis and alternative hypothesis: In hypothesis testing, researchers formulate two competing hypotheses, known as the null hypothesis (H0) and the alternative hypothesis (Ha).

The null hypothesis is the default position or assumption, which states that there is no significant difference or relationship between variables in the population. It represents the belief that any observed differences or relationships in the sample are due to chance. Researchers aim to challenge or reject the null hypothesis in favor of the alternative hypothesis.

The alternative hypothesis, on the other hand, asserts that there is a significant difference or relationship between the variables in the population. It represents what the researchers actually want to prove or show evidence for in their study.

Hypothesis testing involves collecting data from a sample and analyzing it to determine whether the evidence supports rejecting the null hypothesis and accepting the alternative hypothesis. Statistical tests are conducted to assess the likelihood of observing the data if the null hypothesis were true. If the evidence strongly contradicts the null hypothesis, researchers reject it in favor of the alternative hypothesis.

Lastly, let's address the third question:

3. Standard Error: The standard error is a measure of the variability or dispersion of sample means around the population mean. It quantifies the degree of uncertainty or estimation error involved when using sample means to make inferences about the population mean.

The standard error is important in research using sample distributions because it helps provide an estimate of how well the sample mean represents the population mean. A smaller standard error suggests that the sample mean is a more reliable estimate of the population mean.

In the given scenario, we are comparing the standard error for two different samples. The standard error decreases as the sample size increases. Therefore, the standard error for the sample with n = 100 (2 points) would be smaller than the standard error for the sample with n = 16 (5 points). This means that the sample mean of n = 100 is expected to be closer to the population mean compared to the sample mean of n = 16.

In summary, understanding concepts like standard deviation, hypothesis testing (including null and alternative hypotheses), and standard error is crucial in research to gather meaningful insights, make accurate inferences, and measure the variability and precision of sample data.