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Exam Grades The final exam grades for a sample of daytime statistics students and evening statistics students at one college are reported. The classes had the same instructor, covered the same material, and had similar exams. Using graphical and numerical summaries, write a brief description about how grades differ for these two groups. Then carry out a hypothesis test to determine whether the mean grades are significantly different for evening and daytime students. Assume that conditions for a \(t\) -test hold. Select your significance level. Daytime grades: \(100,100,93,76,86,72.5,82,63,59.5,53,79.5\), \(67,48,42.5,39\) Evening grades: \(100,98,95,91.5,104.5,94,86,84.5,73,92.5\), \(86.5,73.5,87,72.5,82,68.5,64.5,90.75,66.5\)

Step by step solution

01

Calculate the means

First, add up all the grades for each group and divide by the number of students in each group to get the mean. For the daytime students, calculate as follows: \((100+100+93+76+86+72.5+82+63+59.5+53+79.5+67+48+42.5+39)/15 \) This gives a mean of 69.5. For the evening students, calculate as follows: \((100+98+95+91.5+104.5+94+86+84.5+73+92.5+86.5+73.5+87+72.5+82+68.5+64.5+90.75+66.5)/19 \) This gives a mean of 84.2.

02

Perform a t-test

After getting the mean of the two data sets, a t-test can be done to see if the means are significantly different. For a one-tailed t-test, set a significance level (for example, \(\alpha = 0.05\) ), then calculate the t-statistic and the degrees of freedom. The formula for the t-statistic is \(\frac{\bar{X}_{1}-\bar{X}_{2}}{\sqrt{\frac{S_{1}^{2}}{n_{1}}+\frac{S_{2}^{2}}{n_{2}}}}\) where \(\bar{X}_{1}\) and \(\bar{X}_{2}\) are the sample means, \(S_{1}^{2}\) and \(S_{2}^{2}\) are the sample variances, and \(n_{1}\) and \(n_{2}\) are the sizes of the two samples.

03

Interpret the results

Finally, compare your calculated t-statistic to the critical t-value from the t-table, based on your chosen significance level and the degrees of freedom. If your calculated t-statistic is greater than the critical t-value, then you can reject the null hypothesis that the means are the same. However, if your calculated t-statistic is less than the critical t-value, you cannot reject the null hypothesis.

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Key Concepts

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

Understanding the t-test

The t-test is a statistical method used to determine if there is a significant difference between the means of two groups. It is particularly useful when dealing with small sample sizes and an unknown population variance. The idea is to compare the average from two different groups and see if they are statistically different from each other.

• Step 1: Mean Calculation — Before you perform a t-test, you need to calculate the mean (average) for both groups. For the daytime students, the mean is 69.5, and for the evening students, it is 84.2.
• Step 2: Calculate the T-Statistic — The t-statistic is calculated using the formula: \( \frac{\bar{X}_{1}-\bar{X}_{2}}{\sqrt{\frac{S_{1}^{2}}{n_{1}}+\frac{S_{2}^{2}}{n_{2}}}} \). This statistic helps determine how far apart the group means are, relative to the spread or variability of the data (standard deviation and variance).

A larger t-statistic indicates a greater difference between the groups. Once calculated, the t-statistic is compared with a critical value from the t-distribution to decide whether the observed difference is significant.

Comparing Means

Mean comparison involves looking at the averages of two groups to check for differences. It is a fundamental part of statistical analysis and gives insight into the central tendency of the data.

• What is a Mean? — A mean is the average of a set of numbers, found by adding all the numbers together and then dividing by the number of values.
• Why Compare Means? — Comparing means can reveal differences in performance, characteristics, or outcomes between groups. In this context, it assesses how well evening students performed compared to daytime students.
• Using Graphs and Tables — Often, graphical summaries like histograms or box plots are used to visually compare means and understand the distribution and spread of data values.

Understanding the mean comparison is crucial for hypothesis testing, as it stimulates curiosity about why differences might exist and whether they are due to chance.

Concept of Significance Level

The significance level, typically denoted by \( \alpha \), is a threshold that determines when an outcome is considered statistically significant. It is a central concept when testing hypotheses, and it helps in deciding whether to reject a null hypothesis.

• Setting the Significance Level — Common levels are 0.05, 0.01, and 0.10, indicating a 5%, 1%, or 10% risk of concluding that a difference exists when there is none. For this exercise, you might use \( \alpha = 0.05 \) as a balanced choice for significance.
• Interpreting the Results — After calculating the t-statistic, compare it to the critical value. If your t-statistic exceeds the critical value, the result is significant, suggesting that the difference in means is unlikely due to random chance.
• Degrees of Freedom — This is related to the number of independent values in your data. It's important in assessing the t-distribution to get accurate critical values based on sample sizes.
  

In hypothesis testing, the significance level is your guide for accepting or rejecting a hypothesis based on your data analysis, ensuring that decisions are scientifically sound and not random.