91Ó°ÊÓ

Exercises 59 to 60 refer to the following setting. For their final project, a group of AP \(^{\otimes}\) Statistics students investigated the following question: "Will changing the rating scale on a survey affect how people answer the question?" To find out, the group took an SRS of 50 students from an alphabetical roster of the school's just over 1000 students. The first 22 students chosen were asked to rate the cafeteria food on a scale of 1 (terrible) to 5 (excellent). The remaining 28 students were asked to rate the cafeteria food on a scale of 0 (terrible) to 4 (excellent). Here are the data: $$ \begin{array}{lcccrc} &{1 \text { to 5 scale }} \\ \text { Rating } & 1 & 2 & 3 & 4 & 5 \\ \text { Frequency } & 2 & 3 & 1 & 13 & 3 \\ \hline & {0 \text { to 4 scale }} \\ \text { Rating } & 0 & 1 & 2 & 3 & 4 \\ \text { Frequency } & 0 & 0 & 2 & 18 & 8 \\ \hline \end{array} $$ Average ratings (1.3,10.2) The students decided to compare the average ratings of the cafeteria food on the two scales. (a) Find the mean and standard deviation of the ratings for the students who were given the 1 -to- 5 scale. (b) For the students who were given the 0 -to- 4 scale, the ratings have a mean of 3.21 and a standard deviation of \(0.568 .\) Since the scales differ by one point, the group decided to add 1 to each of these ratings. What are the mean and standard deviation of the adjusted ratings? (c) Would it be appropriate to compare the means from parts (a) and (b) using a two-sample \(t\) test? Justify your answer.

Step by step solution

01

Calculate Mean for 1 to 5 Scale

To find the mean for the 1 to 5 scale, multiply each rating by its frequency, sum these values, and divide by the total number of ratings.\[\text{Mean} = \frac{(1\times2) + (2\times3) + (3\times1) + (4\times13) + (5\times3)}{22} = \frac{71}{22} \approx 3.23\]

02

Calculate Variance for 1 to 5 Scale

First find the squared differences from the mean for each rating, multiply each by its frequency, sum these, and then divide by the number of ratings.\[\text{Variance} = \frac{(1-3.23)^2\times2 + (2-3.23)^2\times3 + (3-3.23)^2\times1 + (4-3.23)^2\times13 + (5-3.23)^2\times3}{22}\approx 1.226\]

03

Calculate Standard Deviation for 1 to 5 Scale

The standard deviation is the square root of the variance calculated in Step 2.\[\text{SD} = \sqrt{1.226} \approx 1.11\]

04

Adjust Mean for 0 to 4 Scale

Since the decision was to add 1 to each of the ratings from the 0 to 4 scale, the mean will also increase by 1. Thus, the adjusted mean is:\[\text{Adjusted Mean} = 3.21 + 1 = 4.21\]

05

Adjust Standard Deviation for 0 to 4 Scale

Adding a constant to each observation does not change the standard deviation, so the adjusted standard deviation remains the same as the original:\[\text{Adjusted SD} = 0.568\]

06

Evaluate Appropriateness of Two-Sample t-Test

The two sample t-test can be used when comparing means from independent samples, especially when the sample sizes are large enough to assume normality. Here, both samples have more than 20 observations, but we need to ensure the conditions of normality and equal variance are met.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean and Standard Deviation

Understanding the concepts of mean and standard deviation is crucial in statistics. The mean, often called the average, is the sum of all the data points divided by the number of data points. It's a measure of the central tendency of your data, meaning it represents the typical value. To calculate the mean of ratings on a 1 to 5 scale, you take each rating, multiply it by its frequency, and divide by the total number of ratings. This gives you the mean rating for that group, providing a single number that summarizes the entire set of data. The standard deviation, on the other hand, measures the amount of variation or dispersion in your data. A low standard deviation means the data points are close to the mean, while a high standard deviation indicates they are spread out over a wider range. You find the standard deviation by first calculating the variance, which involves finding the squared differences from the mean, multiplying by their frequencies, and then averaging these values. The standard deviation is the square root of this variance.

Two-Sample t-Test

The two-sample t-test is a method used to determine if two populations have different means. This test helps decide whether the difference in means between two samples is significant. For the t-test to be appropriate, several conditions need to be met:

• Independence: The samples must be independent of each other.
• Normality: The data should be approximately normally distributed, or the sample sizes should be large enough (usually n > 30) to rely on the Central Limit Theorem.
• Equal variances: The variance within each sample should be roughly equal, although this isn't strictly required with some variations of the test.

In the exercise, these conditions were evaluated to decide if the test could be applied to compare the means of the two groups surveyed with different scales. This involved considering the sample sizes and distributions to ensure the method was suitable.

Sampling Methods

Sampling methods are techniques used to select a group of subjects or items from a larger population. The goal is to obtain a sample that is representative of the larger group. One popular method is Simple Random Sampling (SRS), where every member of the population has an equal chance of being selected. This method helps to avoid bias in your sample results. For example, selecting students alphabetically from a roster ensures that each student has an equal opportunity to be chosen, leading to results that more accurately reflect the population as a whole. Understanding sampling methods is vital in statistics because the way a sample is collected can significantly affect the validity of the conclusions drawn from it. Proper sampling techniques help ensure that the results of your analysis are reliable and can be generalized to the entire population.

Survey Design

Survey design involves creating a set of questions to gather data from respondents. A well-designed survey should yield clear, unbiased results that answer the research question. Key factors to consider in survey design include:

• Question wording: Questions should be neutral and easy to understand, avoiding leading or loaded phrasing.
• Scale: Choosing the right scale for responses is crucial, as the exercise showed with the 1 to 5 vs. 0 to 4 scales. Different scales can influence how respondents perceive and answer questions.
• Sample size: The number of people surveyed should be enough to provide statistically significant results, yet manageable within logistical constraints.

By carefully tailoring a survey design, researchers can effectively study population characteristics and achieve a deeper understanding of the data gathered.