12D InClass

Question 1

1) Do you think that the font used in printed instructions influences how difficult people think it will be to follow those instructions? Would you rather read a set of instructions printed in Arial font or Mistral font (the fonts are shown below)?

In previous lessons, you constructed confidence interval estimates for a population  proportion, a difference in proportions, and a population mean. The form of those confidence intervals was:

estimate ± margin of error

When you are interested in estimating a difference in population means using data from independent samples, the confidence interval has the same form. The estimate used to construct the interval is the difference in sample means, [latex]\bar{x}_{1}-\bar{x}_{2}[/latex], and the margin of error is calculated using the standard error for a difference in sample means and a critical value from the t Distribution. You will use technology to calculate the margin of error, so we won’t worry about the formula for standard error or margin of error at this point (you  will see them in In-Class Activity 13.C).

Compared to the formula for proportions, the margin of error here is calculated a little differently—instead of multiplying the value of the standard error by a value from the normal distribution, it is multiplied by a value from the appropriate t Distribution. This is not surprising if you think back to your work with the standardized t-statistic in In-Class Activity 12.B.

In the preview assignment, you used data from a study that investigated the effect of using a cell phone while driving on reaction time.[1] In this study, 64 students were randomly assigned to one of two groups. Students in both groups were asked to drive in a driving simulator and to press a brake button as quickly as possible when they saw a red light. Response times (in milliseconds) were measured. Students in one group used their cell phones while driving in the simulator and students in the other group did not use their cell phones. This dataset is built into the data analysis tool, and you will use these data to estimate the difference between the mean reaction time for people who use their cell phones while driving and the mean reaction time for people who do not use their cell phones while driving.

Recall from your previous work that when you want to use a confidence interval, you follow a process that includes deciding what confidence interval method might be used, checking the assumptions for the chosen method to make sure it is appropriate to use,  calculating the confidence interval, and interpreting the interval in context.

Key Takeaways

2) In the preview assignment, the two-sample t confidence interval was selected, and  you verified that the necessary assumptions were reasonable. Now, we will look at  calculating the confidence interval and interpreting it in context.

a) Go to the DCMP Comparing Two Population Means tool at  https://dcmathpathways.shinyapps.io/2sample_mean/. You will use this tool  to calculate confidence intervals for a difference in population means.

Under the Confidence and Significance Tests tab:

  • Select the Reaction Times dataset.
  • For Type of Inference, select “Confidence Interval.”
  • Use the slider for the confidence level to select a 95% confidence level.

Based on the output, what is the 95% confidence interval for the difference in mean  reaction times between people who use their cell phones while driving and people  who do not use their cell phones while driving?

b) Think about how you have previously interpreted confidence intervals for a  difference. How would you interpret the confidence interval you just  calculated?

c) What does the confidence interval tell you about the effect of using a cell  phone while driving on reaction time?

Hint: Both endpoints of the interval are positive. What does this tell you about  the mean reaction times?

d) Which of the following is a correct interpretation of the confidence level of  95%?

(a) The probability that the actual value of the difference in population means between 12.3 and 90.9 milliseconds is 0.95.

(b) 95% of people using their cell phones while driving have a reaction time between 12.3 and 90.9 milliseconds.

(c) If this method was used to construct a confidence interval for the  difference in population means for many different pairs of samples from  the population, about 95% of the intervals would contain the actual  difference in population means.

(d) The mean difference in mean reaction times is guaranteed to be in the  interval from 12.3 to 90.9 milliseconds.

e) If you were to use this sample to calculate a 90% confidence interval rather  than a 95% confidence interval, how would the width of the two intervals  compare?

(a) The width of the two intervals would be the same.

(b) The 95% confidence interval will be wider than the 90% confidence  interval.

(c) The 95% confidence interval will be narrower than the 90% confidence  interval.

As was the case with previous inference methods, there are a few assumptions/conditions that you should check before using the two-sample t confidence interval. These were introduced in the preview assignment and are included here as a  reminder:

  1. The samples are independent.
  2. Each sample is a random sample from the corresponding population of interest or  it is reasonable to regard the sample as if it were a random sample. It is  reasonable to regard the sample as a random sample if it was selected in a way  that should result in the sample being representative of the population. If the data  are from an experiment, you just need to check that there was random  assignment to experimental groups—this substitutes for the random sample  condition and also results in independent samples.
  3. For each population, the distribution of the variable that was measured is  approximately normal, or the sample size for the sample from that population is  large. Usually, a sample of size 30 or more is considered to be “large.” If a sample  size is less than 30, you should look at a plot of the data from that sample (a  dotplot, a boxplot, or, if the sample size isn’t really small, a histogram) to make  sure that the distribution looks approximately symmetric and that there are no  outliers.

Question 3

3) Researchers at the University of Michigan[2] investigated whether the font used in  printed instructions influences how difficult people think it will be to complete a task.  Participants were randomly assigned to one of two groups. Those in one group read  instructions on how to make a Japanese sushi roll that were printed in an easy-to read font (Arial). Participants in the second group read the same instructions, but they were printed in a hard-to-read font (Mistral). All participants were then asked to  say how long they thought it would take to prepare the sushi roll according to the  given instructions.

The mean estimated time to prepare the sushi roll for the sample who read the  instructions in Arial font was 22.71 minutes and the sample standard deviation was  13.76 minutes. The mean estimated time to prepare the sushi roll for the other  sample who read the instructions in Mistral font was 36.15 minutes and the sample  standard deviation was 15.30 minutes. For the purposes of this example, suppose  that the two sample sizes were both 34.

a) Is the two-sample t interval an appropriate way to estimate the difference in  mean estimated times to prepare the sushi roll between those who read the  instructions in the easy-to-read font and those who read the instructions in  the hard-to-read font?

Hint: Are the assumptions for the two-sample t confidence interval  reasonably met?

b) Use the data analysis tool to calculate a 90% confidence interval for the  difference in mean estimated times to prepare the sushi roll between those  who read the instructions in the easy-to-read font and those who read the  instructions in the hard-to-read font.

Go to the DCMP Comparing Two Population Means tool at

https://dcmathpathways.shinyapps.io/2sample_mean/.

Under the Confidence and Significance Tests tab:

  • Select “Summary Statistics” from the drop-down menu under “Enter  Data.”
  • Type in “Estimated Time to Complete” for the name of the variable, and type in “Easy to Read” for the Group 1 label and “Hard to Read” for the  Group 2 label.
  • Enter the sample sizes and the sample means and standard deviations for this example.
  • For Type of Inference, select “Confidence Interval.”
  • Use the slider for the confidence level to select a 90% confidence level.

c) Interpret the confidence interval from Part B.

d) What does the confidence interval tell you about whether the font used in the printed instructions influences how difficult people think it will be to complete a task?

Hint: Both endpoints of the interval are negative. What does this tell you about the difference in mean estimated times to complete the task?

e) What recommendation would you make to a company who provides printed instructions with their products based on your analysis of the data from this  experiment?

  1. Strayer, D. L., & Johnston, W. A. (2001, November 1). Driven to distraction: Dual-task studies of  simulated driving and conversing on a cellular telephone. Psychological Science, 12(6), 462–466.  https://doi.org/10.1111/1467-9280.00386 ↵
  2. Song, H., & Schwarz, N. (2008, October 1). If it's hard to read, it's hard to do: Processing fluency affects  effort prediction and motivation. Psychological Science, 19(10), 986–988. https://doi.org/10.1111/j.1467- 9280.2008.02189.x  ↵