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Students in a representative sample of 65 first-year students selected from a large university in England participated in a study of academic procrastination ("Study Goals and Procrastination Tendencies at Different Stages of the Undergraduate Degree," Studies in Higher Education [2016]: 2028-2043). Each student in the sample completed the Tuckman Procrastination Scale, which measures procrastination tendencies. Scores on this scale can range from 16 to \(64,\) with scores over 40 indicating higher levels of procrastination. For the 65 first-year students in this study, the mean score on the procrastination scale was 37.02 and the standard deviation was 6.44 . a. Construct a \(95 \%\) confidence interval estimate of \(\mu,\) the mean procrastination scale for first-year students at this college. (Hint: See Example 12.7.) b. Based on your interval, is 40 a plausible value for the population mean score? What does this imply about the population of first-year students?

Step by step solution

01

Identify the given information

We are given the following information: - Sample size (n) = 65 - Sample mean (\(\bar{x}\)) = 37.02 - Sample standard deviation (s) = 6.44 - Confidence level = 95%

02

Find the critical value (t) for a 95% confidence interval

In order to construct the confidence interval, we need to find the critical value corresponding to the given confidence level. We will use the t-distribution, since the population standard deviation is unknown. The t-distribution has n-1 degrees of freedom. In this case, we have 65 - 1 = 64 degrees of freedom. To find the critical t-value, we will use a t-table or a calculator that can compute the inverse t-distribution. For a 95% confidence interval with 64 degrees of freedom, the critical t-value (t) is approximately 1.999.

03

Calculate the margin of error

To calculate the margin of error, we will use the following formula: Margin of error = t × (s / √n) Substitute the values to get: Margin of error = 1.999 × (6.44 / √65) Margin of error ≈ 1.569

04

Construct the 95% confidence interval

To construct the confidence interval, we will add and subtract the margin of error from the sample mean: Lower limit = \(\bar{x}\) - Margin of error = 37.02 - 1.569 ≈ 35.45 Upper limit = \(\bar{x}\) + Margin of error = 37.02 + 1.569 ≈ 38.59 So, the 95% confidence interval for the population mean is (35.45, 38.59).

05

Determine if 40 is a plausible value for the population mean

To determine if 40 is a plausible value for the population mean, we need to check if it lies within the calculated confidence interval. Since 40 is outside the interval (35.45, 38.59), it is not considered a plausible value for the population mean.

06

Discuss the implications for the population of first-year students

Since 40 is not a plausible value for the population mean, we can conclude that, on average, first-year students at this college have a lower level of procrastination than the threshold for higher levels of procrastination (40). It indicates that the majority of first-year students may not be categorized as having higher levels of procrastination based on the Tuckman Procrastination Scale.

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Key Concepts

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

Tuckman Procrastination Scale

The Tuckman Procrastination Scale is a psychological assessment tool designed to measure an individual's tendency to procrastinate. Developed by Bruce Tuckman, this self-report questionnaire consists of a series of statements to which respondents indicate their level of agreement. The scale typically ranges from 16 to 64, with higher scores indicating a greater propensity for procrastination.

In academic settings, the scale can reveal important insights into student behavior, particularly in how they approach their studies and manage time. Scores exceeding 40 suggest the student may be prone to putting off tasks, leading to potential academic difficulties or stress. This data is valuable for educators and counselors who aim to develop strategies to help students improve their time management and study skills.

Academic Procrastination

Academic procrastination refers to the act of delaying or postponing academic tasks such as studying, completing assignments, or preparing for examinations. It is a widespread phenomenon that affects a significant number of students at various educational levels. The causes of academic procrastination can vary, including fear of failure, a lack of motivation, or poor time management skills.

Understanding procrastination can lead to better academic support systems and interventions design to prevent procrastination-related academic issues. Further, recognizing the factors contributing to procrastination can help students develop more effective personal strategies for managing their academic workload.

Sample Standard Deviation

The sample standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the data points tend to be close to the mean (or expected value) of the set, while a high standard deviation indicates that the data points are spread out over a larger range of values.

In the context of the Tuckman Procrastination Scale, the standard deviation can give insights into how varied the procrastination tendencies are among the sample of students. A higher standard deviation suggests that students' scores are more diverse, while a lower standard deviation implies that most students have similar levels of procrastination.

T-distribution

The t-distribution, also known as the Student's t-distribution, is a type of probability distribution that is symmetric and bell-shaped but has heavier tails than the normal distribution. It arises when estimates of the population mean are made from a small sample size and the population's standard deviation is unknown.

In constructing confidence intervals, as seen with the Tuckman Procrastination Scale example, the t-distribution is used to determine the critical value or t-score when the sample size is not large enough to assure a normal distribution of the sample mean. This t-score is then utilized to calculate the margin of error and construct a confidence interval that estimates the true population mean. With a given confidence level, you can infer that, with a certain degree of certainty, the population mean will fall within this interval, assuming the population from which the sample is drawn approximates a normal distribution.