# One sample z-test

## What is One Sample z-test?

One Sample z-test is a statistical hypothesis test used to determine whether a sample mean is significantly different from a known population mean, when the population standard deviation is known.

The test is called a "one sample" test because it involves comparing a single sample of data to a known population parameter. The "z-test" part of the name refers to the use of the standard normal distribution to calculate the test statistic.

## When to use One Sample z-test?

One sample z-test is used to test the difference between the sample mean and the population mean when the population standard deviation is known. It is generally used when you have a single sample and want to compare it to a known population value.

Here are some scenarios where you might want to use a one sample z-test:

• Quality control: A factory produces widgets, and you want to know if the weight of the widgets is consistent with the specification of the customer. In this case, you can take a sample of widgets, measure their weight, and compare it to the customer's specification.
• Medical research: A new drug has been developed to lower blood pressure, and you want to know if it is effective. You can take a sample of patients with high blood pressure, give them the drug, and measure their blood pressure after a certain period of time. You can then compare the average blood pressure of the sample to the population mean.
• Customer satisfaction: A company wants to know if its customers are satisfied with its products. In this case, the company can take a random sample of customers and ask them to rate their satisfaction on a scale of 1 to 10. The company can then compare the average satisfaction score of the sample to the population mean.

In general, one sample z-tests are useful when you have a single sample and want to compare it to a known population value. They are commonly used in quality control, medical research, and market research.

## Guidelines for correct usage of One sample z-test

• Use One sample z if the population standard deviation is known
• Use continuous data for analysis, such as weights of packages
• Use One sample Poisson Rate for count data and One Proportion for pass/fail type data
• Sample size should be greater than 20 and data should not be severely skewed
• Select the sample data randomly and ensure each observation is independent
• Determine an appropriate sample size to ensure precise estimates and narrow confidence intervals, and protection errors.

## Alternatives: When not to use One sample z-test

In case you are unaware of the population's standard deviation, you should utilize the One sample t-test.

## Example of One sample z-test?

A scientist for a company that manufactures processed food wants to assess the percentage of fat in the company's bottled sauce. The advertised percentage is 15%. The scientist measures the percentage of fat in 20 random samples. Previous measurements found that the population standard deviation is 2.6%. The scientist performs a One sample z-test to determine whether the fat percentage differs from 15%. She has performed this following steps:

1. She worked all day and gathered the necessary data.

1. Now, she analyzes the data with the help of https://qtools.zometric.com/
2. Inside the tool, she feeds the data. Also, she puts 95 as the confidence level, known standard deviation as 2.6 and hypothesized mean as 15.
3. After using the above mentioned tool, she fetches the output as follows:

## How to do One sample z-test

The guide is as follows:

1. Login in to QTools account with the help of https://qtools.zometric.com/
2. On the home page, you can see One sample z-test under Hypothesis Tests.
3. Click on One sample z-test and reach the dashboard.
4. Next, update the data manually or can completely copy (Ctrl+C) the data from excel sheet and paste (Ctrl+V) it here.
5. Next, you need to put the values of confidence level, known standard deviation and hypothesized mean.
6. Finally, click on calculate at the bottom of the page and you will get desired results.

On the dashboard of One sample z-test, the window is separated into two parts.

On the left part, Data Pane is present. In the Data Pane, each row makes one subgroup. Data can be fed manually or the one can completely copy (Ctrl+C) the data from excel sheet and paste (Ctrl+V) it here.

On the right part, there are many options present as follows:

• Confidence level: In hypothesis testing, the confidence level represents the degree of certainty or level of confidence that we have in our statistical analysis. It is a probability value that indicates the likelihood that the true population parameter falls within the specified range of values.Typically, the confidence level is expressed as a percentage and is denoted by (1 - α), where α is the level of significance or the probability of rejecting a true null hypothesis. For example, if we have a confidence level of 95%, then we are saying that we are 95% confident that the true population parameter lies within our interval estimate, and there is a 5% chance of making a type I error (rejecting a true null hypothesis). In practical terms, a higher confidence level means that we are more confident in our statistical analysis and results. However, increasing the confidence level also increases the width of the confidence interval, making it more difficult to detect small effects. Therefore, the choice of the confidence level depends on the context of the study and the goals of the researcher.
• Known standard deviation: In hypothesis testing, the standard deviation is a measure of the spread of a set of data around its mean. It is used in conjunction with the sample mean to calculate the test statistic, which is then used to determine the p-value of a hypothesis test. The standard deviation of a sample is denoted by the symbol s, and it is calculated as the square root of the sample variance. The sample variance is the average of the squared differences of each observation from the sample mean. In hypothesis testing, the standard deviation is used to determine the significance of the difference between the observed sample mean and the expected population mean under the null hypothesis. Specifically, the test statistic (usually denoted by t) is calculated as the difference between the sample mean and the population mean (assumed under the null hypothesis), divided by the standard error of the mean (which is equal to the standard deviation of the sample divided by the square root of the sample size).
• Hypothesized mean: The hypothesized mean, also known as the expected mean or population mean, is a value specified in the null hypothesis that represents the average value of the variable being tested. It is a theoretical value that is used to compare the actual sample mean to determine whether the difference is statistically significant or due to chance.
• Alternative hypothesis: In hypothesis testing, the alternative hypothesis (also called the research hypothesis) is a statement that represents a different conclusion than the null hypothesis. The null hypothesis typically represents the status quo or the assumption that there is no significant difference or relationship between two or more groups or variables. The alternative hypothesis is the statement that is being tested, and it proposes that there is a significant difference or relationship between the groups or variables being studied.
• Individual value plot: An individual value plot is a type of graphical display that can be used in hypothesis testing to visually examine the distribution of a sample of data and compare it to a null hypothesis distribution. It is also sometimes called a dot plot or dot chart. In an individual value plot, each observation in the sample is represented as a single dot on the graph. The horizontal axis typically represents the values of the variable being measured, and the vertical axis shows the frequency or density of the data.
• Box Plot: A box plot, also known as a box-and-whisker plot, is a graphical representation of data that displays the distribution of a dataset, including its median, quartiles, and any outliers. In hypothesis testing, a box plot can be used to visually compare the distribution of a sample to a known or expected distribution, such as a normal distribution. This can help determine whether the sample data is significantly different from what is expected. The box in a box plot represents the middle 50% of the data, with the lower edge of the box indicating the first quartile (Q1), the upper edge of the box indicating the third quartile (Q3), and the line inside the box indicating the median. The whiskers extend from the box to the minimum and maximum values in the dataset, excluding any outliers, which are plotted as individual points beyond the whiskers.
• Histogram: In hypothesis testing, a histogram is a graphical representation of the distribution of a sample of data. It is a visual tool used to examine the shape and characteristics of the data. A histogram is created by dividing the data into a set of intervals, or bins, and counting the number of data points that fall into each bin. The bins are typically of equal width, and the height of each bar in the histogram represents the frequency of data points that fall within that bin.

## How to do One sample z-test for summarized data

The guide is as follows:

1. Login in to QTools account with the help of https://qtools.zometric.com/
2. On the home page, you can see One sample z-test for summarized data under Hypothesis Tests.
3. Click on One sample z-test for summarized data and reach the dashboard.
4. Next, update the data manually or can completely copy (Ctrl+C) the data from excel sheet and paste (Ctrl+V) it here.
5. Next, you need to put the values of sample size, sample mean, known standard deviation, hypothesized mean and confidence level.
6. Finally, click on calculate at the bottom of the page and you will get desired results.

On the dashboard of One sample z-test for summarized data, the window is having only left part.

In this part, there are many options present as follows:

• Sample size: Sample size refers to the number of individuals, objects, or events selected from a population to be studied in order to draw conclusions about the whole population. In other words, it is the number of observations or participants included in a study. The size of the sample can have a significant impact on the accuracy and reliability of the study's results. A larger sample size typically provides a more representative picture of the population and helps to reduce the effects of random sampling error. Therefore, it is important to determine an appropriate sample size before conducting research to ensure that the results are statistically valid and reliable.
• Sample mean: The sample mean is the average value of a set of observations or data points selected from a larger population. It is calculated by adding up all the values in the sample and dividing by the number of observations. The sample mean is often used as an estimator of the population mean, which is the average value of the entire population.
• Confidence level: In hypothesis testing, the confidence level represents the degree of certainty or level of confidence that we have in our statistical analysis. It is a probability value that indicates the likelihood that the true population parameter falls within the specified range of values. Typically, the confidence level is expressed as a percentage and is denoted by (1 - α), where α is the level of significance or the probability of rejecting a true null hypothesis. For example, if we have a confidence level of 95%, then we are saying that we are 95% confident that the true population parameter lies within our interval estimate, and there is a 5% chance of making a type I error (rejecting a true null hypothesis). In practical terms, a higher confidence level means that we are more confident in our statistical analysis and results. However, increasing the confidence level also increases the width of the confidence interval, making it more difficult to detect small effects. Therefore, the choice of the confidence level depends on the context of the study and the goals of the researcher.
• Known standard deviation: In hypothesis testing, the standard deviation is a measure of the spread of a set of data around its mean. It is used in conjunction with the sample mean to calculate the test statistic, which is then used to determine the p-value of a hypothesis test. The standard deviation of a sample is denoted by the symbol s, and it is calculated as the square root of the sample variance. The sample variance is the average of the squared differences of each observation from the sample mean. In hypothesis testing, the standard deviation is used to determine the significance of the difference between the observed sample mean and the expected population mean under the null hypothesis. Specifically, the test statistic (usually denoted by t) is calculated as the difference between the sample mean and the population mean (assumed under the null hypothesis), divided by the standard error of the mean (which is equal to the standard deviation of the sample divided by the square root of the sample size).
• Hypothesized mean: The hypothesized mean, also known as the expected mean or population mean, is a value specified in the null hypothesis that represents the average value of the variable being tested. It is a theoretical value that is used to compare the actual sample mean to determine whether the difference is statistically significant or due to chance.
• Alternative hypothesis: In hypothesis testing, the alternative hypothesis (also called the research hypothesis) is a statement that represents a different conclusion than the null hypothesis. The null hypothesis typically represents the status quo or the assumption that there is no significant difference or relationship between two or more groups or variables. The alternative hypothesis is the statement that is being tested, and it proposes that there is a significant difference or relationship between the groups or variables being studied.