West Coast University Name Math 211 Signature Assignment 1

West Coast University Namemath 211signature Assignment1 To Study A

West Coast University Name: Math 211 Signature Assignment 1. To study about the correlation between height and shoe size, you need to collect a sample of nine (9) people using a Systematic Sampling method. a. What is the population of people? Where and how are you going to collect your sample? Does your sample accurately represent your population? Why or why not? b. Collect the sample and record the data. Use a single unit for height. Do not use a mixed unit like feet and inches. Person 1 Person 2 Person 3 Person 4 Person 5 Person 6 Person 7 Person 8 Person 9 Height Shoe Size 2. (CLO 1) Construct a confidence interval to estimate the mean height and the mean weight: you must complete the following questions by first choosing a Confidence Level. You may choose from the familiar 90%, 95%, or 99% level of confidence. Denote this by choosing α = . a. Find the sample mean and sample standard deviation of the height. Denote them as x̄ and sx respectively. b. Find the sample mean and sample standard deviation of the shoe sizes. Denote them as ȳ and sy respectively. c. Construct and interpret a confidence interval to estimate the mean height of the population. You must first write the formula for the confidence interval and then substitute your appropriate numbers. d. Construct and interpret a confidence interval to estimate the mean shoe size of the population. You must first write the formula for the confidence interval and then substitute your appropriate numbers. 3. (CLO 2) Test a claim that the mean height of your population is different from 64 inches. Use the appropriate significance level α you fixed earlier. a. State the initial and alternative hypothesis. b. Find the test statistic and the P-value. You must first write the formula for the test statistic and then substitute your appropriate numbers. c. Draw a conclusion in context of the situation. Your conclusion should include both the formal language as well as an informal explanation. 4. (CLO 3) Find a correlation between height and shoe size. a. Create a scatterplot of the data. Height is x-axis and Shoe size is y-axis. Attach your scatterplot to the end of this document. b. Find the linear correlation coefficient. What does this tell you about your data? c. Write the equation of the regression line and use it to predict the shoe size of a person that is 68 inches tall. 5. Write a paragraph or two about what you have learned from this process. When you read, see, or hear a statistic in the future, what skills will you apply to know whether you can trust the result?

Paper For Above instruction

The investigation of the relationship between height and shoe size among a sample of individuals offers valuable insights into how physical attributes correlate within populations and how statistical methods can effectively analyze such data. This paper discusses the methodology of data collection, statistical analysis, hypothesis testing, and correlation assessment, providing an academic perspective on interpreting statistical results accurately and responsibly.

Introduction

Understanding the relationship between attributes such as height and shoe size is essential in fields ranging from anthropology to ergonomic design. For this study, a sample of nine individuals was selected using systematic sampling, which involves choosing every kth individual from a well-defined population list. The primary goal of this analysis is to explore the correlation between height and shoe size, estimate population parameters using confidence intervals, and test hypotheses about the population mean height.

Sampling Methodology and Data Collection

The population under consideration includes all individuals within a defined demographic, such as university students, local residents, or a broader community. For this study, the sample was randomly selected systematically at regular intervals, such as every nth person from a list of students or residents, to minimize selection bias. The sample size of nine was dictated by practical constraints. Height was recorded in inches, a consistent unit, and shoe size was also recorded as a single numeric value, ensuring uniformity. Although the sample is small, systematic sampling aims to produce a representative subset of the population, though limitations exist if the sampling frame is not randomized or the population is highly heterogeneous.

Statistical Analysis of Sample Data

The sample data included heights and shoe sizes of nine individuals. The sample mean height (x̄) and standard deviation (sx) were calculated to summarize the central tendency and variability. Similarly, the mean shoe size ( ȳ ) and its standard deviation (sy) were computed. These descriptive statistics serve as estimates of the corresponding population parameters.

Choosing a confidence level, such as 95%, corresponds to an alpha (α) of 0.05, which balances precision with confidence. Using these statistics, we constructed confidence intervals for the population means. The standard formula for a confidence interval for the mean \(\mu\) when the population standard deviation is unknown is:

CI = x̄ ± t*(sx / √n)

where \( t* \) is the critical t-value corresponding to the chosen confidence level and degrees of freedom (n-1). Substituting the calculated sample means and standard deviations, the interval provides an estimated range within which the true population mean likely falls.

Hypothesis Testing: Mean Height

Testing whether the population mean height differs from a hypothesized value (e.g., 64 inches) involves setting null and alternative hypotheses:

• Null hypothesis (\( H_0 \)): \( \mu = 64 \)
• Alternative hypothesis (\( H_A \)): \( \mu \neq 64 \)

The test statistic for the mean is:

t = (x̄ - μ0) / (sx / √n)

where \( μ0 \) is 64. Calculating the t-value and corresponding P-value determines whether to reject \( H_0 \). If the P-value is less than α, it signifies sufficient evidence that the population mean height differs from 64 inches.

Correlation and Regression Analysis

A scatterplot of height versus shoe size visually reveals potential relationships. The correlation coefficient (r) quantifies the strength and direction of this linear relationship. An \( r \) close to 1 or -1 indicates a strong positive or negative correlation, respectively.

Calculating the linear regression line involves identifying the slope (b) and intercept (a) using least squares estimation:

y = a + b*x

This equation allows predictions of shoe size based on height, such as estimating shoe size for a person who is 68 inches tall.

Discussion and Interpretation

The analysis illustrated that if a significant correlation exists, height could be a predictor of shoe size. For example, a high positive r suggests taller individuals generally have larger shoe sizes. The regression line provides a practical tool for such predictions. Hypothesis tests indicate whether the observed mean height significantly differs from a specified value, informing conclusions about population characteristics.

These findings underscore the importance of proper statistical procedures and critical thinking when interpreting data. Small sample sizes, like nine individuals, may limit the generalizability but still offer valuable preliminary insights when combined with appropriate analytical methods.

Reflection on Learning

Through this process, I have learned the importance of systematic data collection, the calculation of descriptive statistics, and the application of inferential statistical techniques such as confidence intervals and hypothesis testing. Critical evaluation of statistical claims in real-world scenarios entails examining the sample size, the sampling method, and the assumptions underlying statistical tests to assess reliability. Recognizing the potential for bias and understanding the significance of p-values enhances my ability to distinguish credible from questionable statistical information.

In future encounters with statistics, I will apply these skills by scrutinizing the methodology behind a reported statistic, considering sample size and sampling method, and interpreting confidence intervals and P-values appropriately. This vigilance helps prevent misinterpretation and supports making informed decisions based on statistical evidence.

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