91Ó°ÊÓ

Refer to the following setting. Do students who read more books for pleasure tend to earn higher grades in English? The boxplots below show data from a simple random sample of 79 students at a large high school. Students were classified as light readers if they read fewer than 3 books for pleasure per year. Otherwise, they were classified as heavy readers. Each student's average English grade for the previous two marking periods was converted to a GPA scale where \(A+=4.3\), \(A=4.0, A-=3.7, B+=3.3,\) and so on. Reading and grades (10.2) Summary statistics for the two groups from Minitab are provided below. $$ \begin{array}{cccc} \text { Type of reader }\quad\mathrm{N} & \text { Mean } & \text { StDev } & \text { SE Mean } \\ \text {Heavy}\quad47 & 3.640 & 0.324 & 0.047 \\ \text {Light}\quad 32 & 3.356 & 0.380 & 0.067 \end{array} $$ (a) Explain why it is acceptable to use two-sample \(t\) procedures in this setting. (b) Construct and interpret a \(95 \%\) confidence interval for the difference in the mean English grade for light and heavy readers. (c) Does the interval in part (b) provide convincing evidence that reading more causes a difference in students' English grades? Justify your answer.

Step by step solution

01

Justifying the Use of Two-Sample t-Procedure

In this context, we are comparing the means of two independent groups, heavy and light readers. The two-sample t-procedure is appropriate because the sample sizes (47 for heavy readers and 32 for light readers) are each above 30, which allows us to assume the sampling distribution of the sample mean differences is approximately normal due to the Central Limit Theorem. Additionally, the summary statistics (means, standard deviations) are given, and we assume independent random samples from the given large population of students.

02

Calculating the 95% Confidence Interval

First, we calculate the standard error for the difference between the two means using the formula: \[ SE = \sqrt{\left( \frac{S_1^2}{n_1} \right) + \left( \frac{S_2^2}{n_2} \right)} \]Given values: - Heavy readers: \(S_1 = 0.324, n_1 = 47\)- Light readers: \(S_2 = 0.380, n_2 = 32\)Substitute the values:\[ SE = \sqrt{\left( \frac{0.324^2}{47} \right) + \left( \frac{0.380^2}{32} \right)} = \sqrt{0.002233 + 0.004512} = \sqrt{0.006745} \approx 0.082 \]Next, calculate the difference in means:\[ \Delta \bar{x} = \bar{x}_1 - \bar{x}_2 = 3.640 - 3.356 = 0.284 \]Using a standard t-distribution for 95% confidence (critical value \(t^*\) roughly 2, because \(df \approx 60\)), the confidence interval is:\[ CI = \Delta \bar{x} \pm t^* \times SE = 0.284 \pm 2 \times 0.082 \approx (0.120, 0.448) \]

03

Interpretation of the Confidence Interval

The calculated 95% confidence interval (0.120, 0.448) suggests that we are 95% confident that the true mean difference in GPA between heavy readers and light readers falls between 0.120 and 0.448. This range indicates that on average, heavy readers have a higher GPA than light readers.

04

Evaluating Causation

Although the confidence interval indicates a higher GPA for heavy readers compared to light readers, this does not prove causation; it only shows an association. The study's observational nature and the lack of random assignment prevent establishing causality. Confounding variables might also influence both reading habits and academic performance, so the results do not provide convincing evidence that reading more causes a difference in grades.

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Key Concepts

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

Confidence Interval

A confidence interval provides a range of values that estimates an unknown population parameter. In this exercise, we are estimating the difference in the mean English GPAs between two groups of students: heavy and light readers.
The 95% confidence interval, for instance, is calculated as the sample mean difference plus or minus the margin of error. This interval helps us understand the reliability of our sample data. It tells us, in 95 out of 100 similar samples, the true mean difference would fall within this range.
For this exercise, the calculated 95% confidence interval is (0.120, 0.448). This means we are 95% confident that the true difference in mean GPAs between heavy and light readers lies within this interval. A positive interval suggests that heavy readers tend to have a higher GPA than light readers.

Sampling Distribution

Understanding sampling distribution is crucial when performing hypothesis tests or constructing confidence intervals. It refers to the distribution of sample statistics—such as the mean—in a large number of samples taken from the same population.
In the case of our exercise, the sampling distribution of the sample mean differences can be assumed to be approximately normal. This is achieved through the Central Limit Theorem, which allows us to make such an assumption as long as the sample sizes are sufficiently large, typically above 30.
The mean of the sampling distribution equals the mean difference between heavy and light readers' GPAs, while its standard deviation is called the standard error. This forms the basis for calculating our confidence interval.

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental principle in statistics. It states that when you have a sufficiently large sample size, the distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution.
In this exercise, it supports the use of the two-sample t-procedure. By having more than 30 students in both reader categories (47 heavy and 32 light), we can assume the sampling distribution of the difference in means is approximately normal.
This normality assumption is essential when computing confidence intervals or conducting hypothesis tests. It allows statisticians to use the t-distribution, which adjusts for small sample sizes, to make inferences about larger populations.

GPA Comparison

Comparing GPAs between different groups requires careful analysis. In this exercise, we aim to determine whether heavy readers have different average GPAs than light readers.
The comparison is conducted using the difference in average GPAs alongside its confidence interval. The significant positive difference in the calculated interval (0.120, 0.448) suggests that heavy readers, on average, have higher GPAs compared to light readers.
However, while the confidence interval suggests a difference, it does not prove that increased reading causes higher GPAs. This is because of potential confounding variables and the nature of the data, which is observational rather than experimental. Thus, we conclude that while more reading is associated with higher GPAs, causation cannot be inferred.