91Ó°ÊÓ

Twenty-nine college students, identified as having a positive attitude about Mitt Romney as compared to Barack Obama in the 2012 presidential election, were asked to rate how trustworthy the face of Mitt Romney appeared, as represented in their mental image of Mitt Romney's face. Ratings were on a scale of 0 to 7 , with 0 being "not at all trustworthy" and 7 being "extremely trustworthy." Here are the 29 ratings: \({ }^{22}\) \(\begin{array}{llllllllll}2.6 & 3.2 & 3.7 & 3.3 & 3.4 & 3.6 & 3.7 & 3.8 & 3.9 & 4.1\end{array}\) \(\begin{array}{lllllllllll}4.2 & 4.9 & 5.7 & 4.2 & 3.9 & 3.2 & 4.5 & 5.0 & 5.0 & 4.6\end{array}\) \(\begin{array}{lllllllll}4.6 & 3.9 & 3.9 & 5.3 & 2.8 & 2.6 & 3.0 & 3.3 & 3.7\end{array}\) Twenty-nine college students identified as having a negative attitude about Mitt Romney as compared to Barack Obama in the 2012 presidential election, were also asked to rate how trustworthy the face of Mitt Romney appeared. Here are the 29 ratings: \(\begin{array}{lllllllllll}1.8 & 3.3 & 4.3 & 4.4 & 2.5 & 2.6 & 3.5 & 4.2 & 4.7 & 2.5\end{array}\) \(\begin{array}{llllllllll}2.5 & 3.6 & 3.9 & 3.9 & 4.3 & 4.3 & 3.8 & 3.3 & 2.9 & 1.7\end{array}\) \(\begin{array}{lllllllll}3.3 & 3.3 & 3.9 & 4.3 & 4.1 & 3.8 & 3.3 & 5.3 & 5.4\end{array}\) (a) Do the sample means suggest that there is a difference in the mean trustworthy ratings between the two groups? (b) Make stemplots for both samples. Are there any obvious departures from Normality? (c) Test the hypothesis \(H_{0}: \mu_{1}=\mu_{2}\) against the one-sided alternative that students with a positive attitude rate Mitt Romney more trustworthy than those with a negative attitude. What do you conclude from part (a) and from the result of your test?

Step by step solution

01

Calculate Sample Means for Each Group

For the group with a positive attitude, add all the ratings and divide by the number of students (29) to find the average. Do the same for the group with a negative attitude.

02

Compare Sample Means

Compare the two sample means you calculated. Determine if the mean of the group with a positive attitude is greater than the negative group, suggesting a possible difference.

03

Construct Stemplots for Each Group

Create a stemplot for each group to visualize the distribution of ratings. A stemplot helps identify any skewness or outliers in the data, assessing normality.

04

Evaluate Stemplots for Normality

Examine the stemplots to see if data is symmetrically distributed or if there are any significant departures from normality, such as skewness or outliers.

05

Set Up Hypotheses for the Test

State the null hypothesis: \(H_0: \mu_1 = \mu_2\) (means are equal). The alternative hypothesis is: \(H_a: \mu_1 > \mu_2\) (positive attitude ratings are higher).

06

Conduct Hypothesis Test

Use a t-test to compare the means of the two independent samples. Calculate the test statistic and find the p-value for the one-sided test.

07

Interpret Results of Hypothesis Test

If the p-value is less than the significance level (usually 0.05), reject the null hypothesis, indicating a significant difference in ratings. Otherwise, do not reject the null hypothesis.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Sample Means

In statistics, a sample mean is the average of all the data points collected in a sample. It serves as an estimate of the population mean, which represents the entire set from which the sample is drawn. In our exercise, we need to calculate the sample means for two groups of students. These groups provide ratings on how trustworthy Mitt Romney's face appeared during the 2012 election period. To compute the sample mean:

• Add all the rating scores within each group.
• Divide this sum by the number of ratings presented (in our case, 29 ratings per group).

After calculating, compare these means. Does the group with a positive attitude towards Mitt Romney rate him as more trustworthy than the group with a negative attitude? Observing such a difference might imply a variance in perception based on attitudes, but you will later need further statistical testing to substantiate this observation.

Stemplots

Stemplots, also known as stem-and-leaf plots, are a fantastic way to quickly visualize the distribution of a dataset. They're simple to construct and allow students to see the shape, central tendency, and spread of the data. In a stemplot, each number in the dataset is split into two parts: the stem, which is one of or more of the leading digits, and the leaf, which is usually the last digit. To create a stemplot:

• Divide each rating number into a stem and a leaf.
• List the stems in a vertical line.
• Attach the leaves to their corresponding stems in increasing order.

This plot provides a clear view of the data, highlighting any peculiarities or trends. It’s also helpful in assessing the normality of the dataset. Through the stemplot, one can spot outliers or detect any skewness in the data distribution at a glance.

Normality Assessment

Normality assessment involves checking whether the data follows a normal distribution, which is a bell-shaped, symmetric distribution prevalent in nature and statistics. In observational studies, verifying normality is a crucial step because many statistical tests, like the t-test, assume that the data are normally distributed. When reviewing stemplots for normality, observe:

• If the plot appears roughly symmetric around the center.
• If the tails on both ends are evenly distributed.
• For significant skews or outliers that could suggest deviations from normality.

Departure from normality can affect the results of hypothesis testing, potentially rendering them unreliable. Thus, before progressing with a t-test, ensure that any skewness is reasonable or apply necessary transformations to the data.

t-test

The t-test is a statistical test used to determine if there is a significant difference between the means of two groups. It's highly utilized in hypothesis testing to compare group means and assess whether observed differences are statistically meaningful or simply due to random chance. Here’s a simplified guide to conducting a t-test:

• Establish your null hypothesis (\(H_0: \mu_1 = \mu_2\)) and alternate hypothesis (\(H_a: \mu_1 > \mu_2\)).
• Calculate the t-statistic using the formula given for comparing means of two independent samples.
• Find the p-value associated with the calculated t-statistic for the one-sided test.
• Compare the p-value to the significance level (typically 0.05).

If the p-value is lower than the significance level, reject the null hypothesis, indicating a significant difference between groups. Otherwise, there isn’t enough evidence to conclude a difference exists. This process helps you derive insights about the data, confirming or challenging your initial observations.