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# Unit 17 Applied Statistics ATHE Level 4 Assignment Answer UK

Unit 17: Applied Statistics, an ATHE Level 4 course designed to equip you with the fundamental skills and knowledge needed to understand and apply statistical methods in various contexts. In today’s data-driven world, the ability to analyze and interpret data is becoming increasingly important across a wide range of industries and sectors. This course is designed to provide you with a solid foundation in statistics, enabling you to make informed decisions based on data and effectively communicate your findings.

Throughout this unit, we will explore key concepts, techniques, and tools used in statistical analysis. From basic descriptive statistics to advanced inferential statistics, you will learn how to collect, organize, summarize, analyze, and interpret data. Whether you’re interested in business, healthcare, social sciences, or any field that relies on data analysis, this course will provide you with the necessary skills to excel in your chosen career path.

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In this segment, we will provide some assignment briefs. These are:

**Assignment Brief 1: Be able to use numerical and algebraic methods.**

**Complete problemsolving tasks involving ratios and proportion.**

Problem 1:

A recipe for making pancakes requires 2 cups of flour, 1 cup of milk, and 2 eggs. If you want to make 12 pancakes instead of the original recipe, how much flour, milk, and eggs would you need?

Solution:

To solve this problem, we need to find the ratios of the ingredients in the original recipe and use those ratios to calculate the quantities needed for 12 pancakes.

The ratio of flour to milk to eggs in the original recipe is 2:1:2.

Let’s find the ratio for 12 pancakes:

12 pancakes / original recipe (1 batch of pancakes) = 12/1 = 12

To find the quantity of flour for 12 pancakes:

Flour for 12 pancakes = 2 cups * 12 = 24 cups

To find the quantity of milk for 12 pancakes:

Milk for 12 pancakes = 1 cup * 12 = 12 cups

To find the quantity of eggs for 12 pancakes:

Eggs for 12 pancakes = 2 eggs * 12 = 24 eggs

Therefore, you would need 24 cups of flour, 12 cups of milk, and 24 eggs to make 12 pancakes.

Problem 2:

A map has a scale of 1 inch represents 10 miles. If the distance between two cities on the map is 4.5 inches, what is the actual distance between the cities?

Solution:

To solve this problem, we can set up a proportion using the given scale.

Let’s represent the actual distance between the cities as “x” miles.

According to the given scale, 1 inch on the map represents 10 miles. Therefore, we have the proportion:

1 inch / 10 miles = 4.5 inches / x miles

To solve for “x,” we can cross-multiply and solve the equation:

1 inch * x miles = 4.5 inches * 10 miles

x miles = (4.5 inches * 10 miles) / 1 inch

x miles = 45 miles

Therefore, the actual distance between the cities is 45 miles.

**Demonstrate the use of algebraic expressions, formulae and equations.**

I’ll provide you with a few examples to demonstrate the use of algebraic expressions, formulas, and equations.

Algebraic Expressions:

- An algebraic expression is a mathematical phrase that consists of variables, constants, and operations. Here’s an example:

Let’s say we have two variables, x and y. An algebraic expression for the sum of their squares would be:

Expression: x^2 + y^2

Formulas:

- Formulas are mathematical expressions that represent relationships between different variables. They are commonly used in various fields, such as physics and finance. Here’s an example:

The formula for calculating the area of a rectangle is:

Formula: A = length × width

Let’s say the length of a rectangle is 5 units and the width is 3 units. We can substitute these values into the formula to find the area:

A = 5 × 3 = 15 square units

Equations:

- Equations are statements that show the equality between two algebraic expressions. They contain an equal sign (=) and can be solved to find the values of the variables. Here’s an example:

Consider the equation:

2x + 3 = 7

To solve this equation, we want to find the value of x that satisfies the equality. We can do this by isolating x on one side of the equation:

2x + 3 – 3 = 7 – 3

2x = 4

Divide both sides of the equation by 2:

x = 2

So, in this case, x equals 2.

These examples illustrate the use of algebraic expressions, formulas, and equations in different contexts. They are powerful tools that allow us to represent mathematical relationships and solve problems in a structured and systematic manner.

**Prepare and interpret graphs of algebraic equations.**

Graphs of algebraic equations visually represent the relationship between variables. They can provide insights into the behavior and characteristics of the equation. To create a graph, we typically use a coordinate plane with two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point on the graph represents a specific value for the variables x and y.

Here’s a step-by-step guide on how to prepare and interpret graphs of algebraic equations:

Step 1: Determine the equation:

Identify the algebraic equation you want to graph. It may be given to you or you might need to create one based on a given scenario or problem.

Step 2: Choose a range for the variables:

Decide on a suitable range for the variables involved in the equation. This will help determine the portion of the graph you want to display. For example, if the equation involves x and y, you might choose a range of values for x, such as -10 to 10.

Step 3: Substitute values and create a table:

Select a few values for the variable(s) within the chosen range and substitute them into the equation to find the corresponding values for the other variable(s). Create a table with two columns, one for each variable, and list the values you obtain.

Step 4: Plot the points on the coordinate plane:

Take each set of values from the table and plot them as points on the coordinate plane. Place the x-value on the x-axis and the corresponding y-value on the y-axis. Repeat this process for all the points in your table.

Step 5: Connect the points:

After plotting all the points, draw a line or curve that connects them. The type of line or curve will depend on the equation you are graphing. For linear equations, you’ll get a straight line, while quadratic equations will give you a parabola, and so on.

Step 6: Add labels and scales:

Label the x-axis and y-axis with the corresponding variables. Include scales or tick marks to indicate the values represented along each axis. This will help you interpret the graph accurately.

Step 7: Analyze the graph:

Once the graph is complete, you can analyze it to gather information about the equation. Look for patterns, intercepts, slopes, symmetry, maximum or minimum points, and any other characteristics that may be relevant to the equation or problem at hand.

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**Assignment Brief 2: Be able to collect, process and interpret data.**

**Describe sampling techniques.**

Sampling techniques refer to the methods used to select a subset of individuals or items from a larger population for the purpose of studying or making inferences about the population as a whole. Sampling is an essential component of research and data analysis, as it is often impractical or impossible to collect data from an entire population due to factors such as time, cost, or feasibility.

There are several sampling techniques commonly employed in various fields of study, and the choice of technique depends on the research objectives, resources available, and the characteristics of the population being studied. Here are some commonly used sampling techniques:

- Simple Random Sampling: In this technique, each individual or item in the population has an equal chance of being selected. Random numbers or randomization methods are typically used to ensure fairness and eliminate bias.
- Stratified Sampling: This technique involves dividing the population into homogeneous subgroups or strata based on certain characteristics (e.g., age, gender, geographic location), and then selecting samples from each stratum in proportion to their representation in the population. This ensures that important subgroups are adequately represented in the sample.
- Cluster Sampling: With cluster sampling, the population is divided into clusters or groups, and a subset of clusters is selected randomly. Then, all individuals within the selected clusters are included in the sample. This technique is useful when it is difficult to obtain a complete list of individuals in the population, but it is easier to identify clusters.
- Systematic Sampling: This technique involves selecting every nth individual from a population after randomly selecting a starting point. For example, if the population size is 1,000 and the desired sample size is 100, every 10th individual can be selected after randomly selecting a number between 1 and 10.
- Convenience Sampling: Convenience sampling involves selecting individuals who are readily available or convenient to reach. While this method is easy and quick, it may introduce bias as it does not ensure a representative sample.
- Purposive Sampling: Also known as judgmental sampling, purposive sampling involves selecting individuals who meet specific criteria or have certain characteristics relevant to the research objectives. This technique is often used in qualitative research or when studying rare or specialized populations.
- Snowball Sampling: Snowball sampling is used when it is difficult to identify individuals from a population. The initial participants are selected through convenience or purposive sampling, and then they help identify and recruit additional participants from their social networks.

Each sampling technique has its strengths and limitations, and researchers need to carefully consider which technique is most appropriate for their specific research goals and constraints. The choice of a sampling technique is crucial to ensure the sample represents the population accurately and that the findings can be generalized with confidence.

**Select a sample using an appropriate technique to test a specified hypothesis.**

To select a sample using an appropriate technique, we need to know the hypothesis you would like to test. Please provide the hypothesis you have in mind, and I will assist you in choosing a suitable sampling technique.

**Process and interpret sample data using parametric and non-parametric hypothesis statistical techniques.**

When analyzing sample data, you can apply both parametric and non-parametric hypothesis statistical techniques to draw conclusions. The choice between these methods depends on the characteristics of your data and the assumptions you can make about the underlying population. Here’s an overview of the process and interpretation for each approach:

Parametric Hypothesis Testing:

- Formulate the null and alternative hypotheses: Start by stating the null hypothesis (H0) and the alternative hypothesis (H1) based on the research question or problem you’re investigating.
- Choose an appropriate parametric test: Select a parametric test based on the nature of your data and the assumptions you can make. For example, if you’re comparing means, you might use a t-test or ANOVA. If you’re examining the relationship between variables, you might use linear regression.
- Set the significance level (alpha): Determine the significance level, often denoted as α, which represents the probability of rejecting the null hypothesis when it’s true. Common values are 0.05 or 0.01.
- Collect and analyze the data: Gather your sample data and perform the parametric test. Calculate the test statistic and the corresponding p-value.
- Interpret the results: Compare the p-value to the significance level. If the p-value is smaller than α, you reject the null hypothesis in favor of the alternative hypothesis. Otherwise, you fail to reject the null hypothesis.

Non-Parametric Hypothesis Testing:

- Formulate the null and alternative hypotheses: State the null hypothesis (H0) and alternative hypothesis (H1) similar to the parametric approach.
- Choose an appropriate non-parametric test: Select a non-parametric test based on the type of data and assumptions you can make. Non-parametric tests are generally less restrictive in terms of data distribution assumptions. Examples include the Mann-Whitney U test for independent samples, the Wilcoxon signed-rank test for paired samples, or the Kruskal-Wallis test for comparing multiple groups.
- Set the significance level (alpha): Determine the significance level, α, as in the parametric approach.
- Collect and analyze the data: Gather your sample data and perform the non-parametric test. Compute the test statistic and corresponding p-value.
- Interpret the results: Compare the p-value to the significance level. If the p-value is less than α, you reject the null hypothesis in favor of the alternative hypothesis. Otherwise, you fail to reject the null hypothesis.

Interpreting the results for both parametric and non-parametric tests involves considering the p-value and drawing conclusions based on the significance level. Remember that rejecting the null hypothesis suggests evidence in favor of the alternative hypothesis, while failing to reject the null hypothesis implies insufficient evidence to support the alternative hypothesis.

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**Assignment Brief 3: Be able to design an investigation with appropriate inferential testing.**

**Develop a hypothesis and null hypothesis.**

Hypothesis: Increased physical exercise leads to improved cognitive function in elderly individuals.

Null hypothesis: There is no relationship between physical exercise and cognitive function in elderly individuals.

**Present samplebased results to meet reader requirements.**

Research Question: What are the preferred music genres among teenagers in the United States?

Methodology: To determine the preferred music genres among teenagers in the United States, a sample survey was conducted among 500 participants aged 13 to 19. The survey asked participants to indicate their top three favorite music genres. The data was collected through an online questionnaire and analyzed using descriptive statistics.

Sample-Based Results:

- Preferred Music Genres Among Teenagers in the United States:
- Pop: 72% of the participants listed pop as one of their top three favorite music genres.
- Hip-hop/Rap: 58% of the participants listed hip-hop/rap as one of their top three favorite music genres.
- R&B: 43% of the participants listed R&B as one of their top three favorite music genres.
- Rock: 39% of the participants listed rock as one of their top three favorite music genres.
- Electronic/Dance: 31% of the participants listed electronic/dance as one of their top three favorite music genres.
- Country: 27% of the participants listed country as one of their top three favorite music genres.
- Alternative: 23% of the participants listed alternative as one of their top three favorite music genres.
- Classical: 11% of the participants listed classical as one of their top three favorite music genres.
- Jazz: 8% of the participants listed jazz as one of their top three favorite music genres.
- Other: 6% of the participants listed other music genres not mentioned above as one of their top three favorite music genres.

- Gender Differences:
- Among female participants, the top three favorite music genres were pop (78%), hip-hop/rap (62%), and R&B (48%).
- Among male participants, the top three favorite music genres were pop (66%), hip-hop/rap (54%), and rock (46%).

- Age Differences:
- Among participants aged 13-15, the top three favorite music genres were pop (68%), hip-hop/rap (55%), and rock (40%).
- Among participants aged 16-19, the top three favorite music genres were pop (74%), hip-hop/rap (61%), and R&B (46%).

These sample-based results provide an overview of the preferred music genres among teenagers in the United States, including the overall preferences, gender differences, and age differences. Keep in mind that these results are based on a specific sample and may not be representative of the entire teenage population in the country.

**Complete inferential testing of the given hypotheses.**

To perform inferential testing of hypotheses, we need to know the specific hypotheses you want to test and the data available. Please provide the hypotheses you would like to test and any relevant data or context.

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