Titleabc123 Version X161 Practice Set 5

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This distribution has only one parameter. The curve is skewed to the right for small degrees of freedom (df) and becomes symmetric for large df. The entire distribution curve lies to the right of the vertical axis, and the distribution assumes nonnegative values only.

Identify the correct distribution: A. t distribution, B. Normal distribution, C. Chi-square distribution, D. Linear regression.

Find the value of chi-square (x²) for 12 degrees of freedom and an area of 0.035 in the right tail of the chi-square distribution curve. Round your answer to three decimal places.

Determine the value of chi-square (x²) for 14 degrees of freedom and an area of 0.25 in the left tail of the chi-square distribution curve. Round your answer to three decimal places.

Determine the value of chi-square (x²) for 23 degrees of freedom and an area of 0.95 in the left tail of the chi-square distribution curve. Round your answer to three decimal places.

Define the test: A. Goodness-of-fit test, B. Chi-square test, C. Linear regression.

The frequencies obtained from performing a multinomial experiment are called the ________, and the frequencies expected if the null hypothesis is true are called the ________.

Fill in the blanks: The observed frequencies are the ________, and the expected frequencies are the ________.

The expected frequency of a category is given by E = np, where n is the sample size and p is the probability that an element belongs to that category if the null hypothesis is true. The ________ for a goodness-of-fit test are k – 1, where k is the number of categories.

Describe a statistical model that involves only two variables, one independent and one dependent. Identify each variable's role in the model.

Given a population data set: N=465, Σx=3920, Σy=2650, Σxy=26,570, Σx²=48,530, compute the population regression line. Round to three decimal places.

From a sample data set: n=12, Σx=66, Σy=588, Σxy=2244, Σx²=396, find the estimated regression line. Round to three decimal places.

Paper For Above instruction

The chi-square distribution is a vital probability distribution used extensively in inferential statistics, particularly in hypothesis testing and goodness-of-fit assessments. It is characterized by its single parameter, degrees of freedom (df), and is known for its skewness to the right, which diminishes as df increases, rendering the distribution more symmetric. This skewness and the distribution's domain—nonnegative values—are key features that distinguish it from other distributions like the normal or t-distribution.

The chi-square distribution's shape is dependent on the degrees of freedom. When df is small, the distribution is notably skewed to the right, indicating that smaller values are more probable, with a long tail extending towards larger values. As df increases, the distribution approaches a more symmetric shape, resembling the normal distribution due to the Central Limit Theorem. This characteristic is essential when selecting critical values for chi-square tests, such as tests for independence or goodness-of-fit, where the area in either tail corresponds to the significance level.

Calculating the critical chi-square (x²) values involves identifying the point at which the area under the distribution curve in a specified tail equals the significance level. For example, for 12 degrees of freedom and a right-tail area of 0.035, one would consult chi-square distribution tables or statistical software to find the corresponding critical value. Similarly, for 14 df with 0.25 area in the left tail and 23 df with 0.95 area in the left tail, the process remains consistent. These critical values are fundamental for hypothesis decision-making—rejecting the null hypothesis if the computed statistic exceeds the critical value (for a right-tail test) or falls below (for a left-tail test).

The chi-square test is central to assessing how well observed data fit a theoretical distribution, known as the goodness-of-fit test. It compares observed frequencies from multinomial experiments with expected frequencies based on a specified pattern. Observed frequencies are obtained directly from experimental data, while expected frequencies are calculated using the model assumptions, such as probability distributions.

The formula for expected frequency, E = np, combines the total sample size n and the probability p of the category under the null hypothesis. For conducting a goodness-of-fit test, the degrees of freedom are determined by subtracting one from the number of categories (k – 1). This adjustment accounts for the constraints imposed on the expected frequencies, such as the total sum equaling n, thereby reducing the parameters used to estimate the distribution.

Modeling relationships between variables is fundamental in statistics. A simple regression model involves one independent variable and one dependent variable. The dependent variable is the one being explained or predicted, whereas the independent variable serves as the predictor or explanatory variable. This model aims to quantify how changes in the independent variable influence the dependent variable, often represented by the equation y = a + bx, where a is the intercept, and b is the slope coefficient.

Given population data: N=465, Σx=3920, Σy=2650, Σxy=26,570, Σx²=48,530, the regression line can be calculated using the formulas for the slope and intercept. The slope (b) is obtained by (NΣxy - ΣxΣy) / (NΣx² - (Σx)²), and the intercept (a) by (Σy - bΣx) / N. Substituting the values yields the regression equation, which explains the relationship between the variables.

Similarly, for the sample data with n=12, Σx=66, Σy=588, Σxy=2244, Σx²=396, the estimated regression line is derived by calculating the slope: b = (nΣxy - ΣxΣy) / (nΣx² - (Σx)²), and the intercept: a = (Σy - bΣx) / n. This estimated line models the relationship based on the sample, providing insights into the predictive relationship between x and y.

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