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- https://stats.libretexts.org/Courses/Lumen_Learning/Concepts_in_Statistics_(Lumen)/03%3A_Examining_Relationships-_Quantitative_Data/3.21%3A_Linear_Relationships_(2_of_4)where n is the sample size; x is a data value for the explanatory variable; is the mean of the x-values; is the standard deviation of the x-values; similarly, for the terms involving y. Use the simula...where n is the sample size; x is a data value for the explanatory variable; is the mean of the x-values; is the standard deviation of the x-values; similarly, for the terms involving y. Use the simulation below to investigate how the value of relates to the direction and strength of the relationship between the two variables in the scatterplot. In the simulation, use the slider bar at the top of the simulation to change the value of the correlation coefficient (r) between −1 and 1.
- https://stats.libretexts.org/Courses/Lumen_Learning/Concepts_in_Statistics_(Lumen)/03%3A_Examining_Relationships-_Quantitative_Data/3.01%3A_Why_It_Matters-_Examining_Relationships-_Quantitative_DataBefore we begin Examining Relationships: Quantitative Data, let’s see how the new ideas in this module relate to what we learned in the previous modules, Types of Statistical Studies and Producing Dat...Before we begin Examining Relationships: Quantitative Data, let’s see how the new ideas in this module relate to what we learned in the previous modules, Types of Statistical Studies and Producing Data and Summarizing Data Graphically and Numerically. Explore the Data: Analyze and summarize the data. ← Summarizing Data Graphically and Numerically, Examining Relationships: Quantitative Data
- https://stats.libretexts.org/Courses/Lumen_Learning/Concepts_in_Statistics_(Lumen)/06%3A_Probability_and_Probability_Distributions/6.24%3A_Introduction_to_Normal_Random_VariablesWhat you’ll learn to do: Use a normal probability distribution to estimate probabilities and identify unusual events. In statistics, the normal random variable is a powerful tool in estimating probabi...What you’ll learn to do: Use a normal probability distribution to estimate probabilities and identify unusual events. In statistics, the normal random variable is a powerful tool in estimating probabilities in hypothesis testing. Many statistical tests will use this standard random variable, so building a solid understanding of how to work with the normal random variable is critical to building up our statistical tool box.
- https://stats.libretexts.org/Courses/Lumen_Learning/Concepts_in_Statistics_(Lumen)/01%3A_Types_of_Statistical_Studies_and_Producing_Data/1.13%3A_Putting_It_Together-_Types_of_Statistical_Studies_and_Producing_DataIn an observational study, we draw a conclusion about the population on the basis of a sample. The size of the population does not affect the accuracy of a random sample as long as the population is l...In an observational study, we draw a conclusion about the population on the basis of a sample. The size of the population does not affect the accuracy of a random sample as long as the population is large. If an attempt is made to include every individual from a population in a sample, then the investigation is called a census. To establish a cause-and-effect relationship, we want to make sure the explanatory variable is the only factor that impacts the response variable.
- https://stats.libretexts.org/Courses/Lumen_Learning/Concepts_in_Statistics_(Lumen)/01%3A_Types_of_Statistical_Studies_and_Producing_Data/1.12%3A_Conducting_Experiments_(2_of_2)At the end of 1 month, the rats in the Mozart group were much faster at running the maze than were the rats in the other two groups. It must be remembered that the WHI data on which the concerns were ...At the end of 1 month, the rats in the Mozart group were much faster at running the maze than were the rats in the other two groups. It must be remembered that the WHI data on which the concerns were raised related to overweight North American women in their mid-sixties and not to the women that are treated with HRT for their menopausal symptoms in the United Kingdom, who are usually around the age of menopause, namely 45–55 years.
- https://stats.libretexts.org/Courses/Lumen_Learning/Concepts_in_Statistics_(Lumen)/04%3A_Nonlinear_Models/4.09%3A_Putting_It_Together-_Nonlinear_ModelsIn this case, the association is positive, and y is increasing. From the growth factor, we can determine the percent increase in y for each additional 1-unit increase in x. In this case, the associati...In this case, the association is positive, and y is increasing. From the growth factor, we can determine the percent increase in y for each additional 1-unit increase in x. In this case, the association is negative, and y is decreasing. From the decay factor, we can determine the percentage decrease in y for each additional 1-unit increase in x. From the slope, we can determine the amount and direction the y-value changes for each additional 1-unit increase in x.
- https://stats.libretexts.org/Courses/Lumen_Learning/Concepts_in_Statistics_(Lumen)/08%3A_Inference_for_One_Proportion
- https://stats.libretexts.org/Courses/Lumen_Learning/Concepts_in_Statistics_(Lumen)/03%3A_Examining_Relationships-_Quantitative_Data/3.08%3A_Putting_It_Together-_Examining_Relationships-_Quantitative_DataWe have two numeric measures to help us judge how well the regression line models the data: The square of the correlation, r 2 , is the proportion of the variation in the response variable that is exp...We have two numeric measures to help us judge how well the regression line models the data: The square of the correlation, r 2 , is the proportion of the variation in the response variable that is explained by the least-squares regression line.
- https://stats.libretexts.org/Courses/Lumen_Learning/Concepts_in_Statistics_(Lumen)/08%3A_Inference_for_One_Proportion/8.08%3A_Hypothesis_Testing_(2_of_5)If the P-value in the previous example was greater than 0.05, then we would not have enough evidence to reject H 0 and accept H a . In this case our conclusion would be that “there is not enough evide...If the P-value in the previous example was greater than 0.05, then we would not have enough evidence to reject H 0 and accept H a . In this case our conclusion would be that “there is not enough evidence to show that the mean amount of data used by teens with smart phones has increased.” Notice that this conclusion answers the original research question.
- https://stats.libretexts.org/Courses/Lumen_Learning/Concepts_in_Statistics_(Lumen)/03%3A_Examining_Relationships-_Quantitative_Data/3.06%3A_Assessing_the_Fit_of_a_Line_(3_of_4)We are looking at 4 years of data, and we see a lot of variation in temperature, so the day of the year only partially explains the increase in temperature. Other variables also influence the temperat...We are looking at 4 years of data, and we see a lot of variation in temperature, so the day of the year only partially explains the increase in temperature. Other variables also influence the temperature, but the line accounts only for the relationship between the day of the year and temperature. The value of r 2 is the proportion of the variation in the response variable that is explained by the least-squares regression line.
- https://stats.libretexts.org/Courses/Lumen_Learning/Concepts_in_Statistics_(Lumen)/10%3A_Inference_for_Means/10.27%3A_Introduction_to_Inference_for_a_Difference_in_Two_Population_MeansWhat you’ll learn to do: Conduct a hypothesis test or construct a confidence interval to investigate a difference between two population means. We will build on this and learn to conduct a hypothesis ...What you’ll learn to do: Conduct a hypothesis test or construct a confidence interval to investigate a difference between two population means. We will build on this and learn to conduct a hypothesis test about a difference between two population means and state a conclusion in context under appropriate conditions. We will then construct a confidence interval to estimate a difference in two population means (when conditions are met) and interpret the confidence interval in context.
