In this activity you will use the IQ.txt data file and StatCrunch to illustrate your understanding of a variety of concepts and skills learned thus far in the course. You should demonstrate proficiency on all homework in chapters 5-7 a before completing this activity. Watch the “StatCrunch Orientation” Talon video to locate the data file.
You will likely find this activity more challenging than the MyLab homework for several reasons. First off, you are working with an actual data set. You will need to spend a bit of time familiarizing yourself with the variables included in the set. Often students rely on the help aids or prompting included in algorithmically generated questions. None of those hints are present in the data set. Additionally, writing a sentence from scratch is more difficult, and more authentic, than selecting the correct word from a dropdown menu. Ask if you have questions. Complete every part of every question to the best of your ability. Expect to resubmit based upon instructor feedback.
I suggest you download this Word document and add your work directly to it. Click into the text box and add your answers to avoid Word renumbering the document. Provide an explanation of your work or thought process to arrive at your solution. Whenever you use StatCrunch, include screen captures of your output to support your answers. StatCrunch output alone is not sufficient – you must interpret the results. Once completed, save as a Word or pdf document and submit to the Dropbox in Talon.
About the data: Researchers want to investigate whether intelligence is associated with income and wealth. They sampled 500 American adults born between 1980 and 1983. The researchers administered an IQ test and computed IQ scores. A person is said to be naturally gifted if their IQ is 125 or above. Researchers also collected gender, 2018 annual income (rounded to nearest $500), and net worth at age 37 (rounded to nearest $10000).
Give an overall summary of the data for IQ including the shape of the distribution and appropriate measures of center and variation. Include a histogram with your summary.
Give an overall summary of the data for Net Worth including the shape of the distribution and appropriate measures of center and variation. Include a histogram with your summary.
Examine the association between Annual Income and Net Worth.
Draw a scatterplot of the data showing the linear regression line with Annual Income as the independent variable and Net Worth as the dependent variable.
State the correlation coefficient. Use StatCrunch to confirm the simple linear regression is N=3.47 I-22073, where N is Net Worth in dollars and I is annual income in dollars.
Interpret the slope of 3.47 in this context.
Interpret the y-intercept of -22073 in this context.
Use the data set to calculate the empirical probability of randomly selecting a volunteer for each of the following situations.
The volunteer selected is female.
The volunteer selected has an Annual Income of at least $100,000.
The volunteer selected is a male and has an IQ above 110.
The volunteer selected has an IQ above 110 given the person selected is a male.
Three volunteers are selected at random, with replacement, from the sample. What is the probability all three are female?
Three volunteers are selected at random, with replacement, from the sample. What is the probability at least one has Net Worth of $1,000,000 or more?
Draw a histogram of the variable IQ. In Display Options, choose to overlay a normal distribution.
Provide a picture to confirm the graph is approximately normal.
State the mean and standard deviation of IQ. Round to one decimal place.
According to the Empirical Rule, 95% of IQ data should fall between what two values?
Open the Normal calculator in StatCrunch (Stat?Calculators?Normal) and choose Between. Then enter the Mean and Standard Deviation from part b. Find the probability, P(Lower=X=Upper), where lower and upper are the values you determined in part c. Include a screen capture. Note: The probability should be very close to .95 because the histogram is approximately normal.
When the distribution is approximately Normal, we can use the Normal calculator to compute theoretical probabilities.
What is the probability of selecting a volunteer with IQ of at least 100?
Compute the z-score for an IQ of 100.
What IQ score is in the 75th percentile? Round to the nearest integer.
What z-score corresponds to the 75th percentile?
Estimating the population proportion of gifted natural intelligence. Before starting this question, confirm the sample proportion of adults with gifted natural intelligence is p ^=0.026.
Suppose we select a new random sample of 500 adults born between 1980 and 1983. Should we expect our sample proportion of people with gifted natural intelligence to be .026? Justify your answer.
Random and Independent
Large Sample
Big Population
Discuss how each condition for the Central Limit Theorem is met by this sample.
Estimate the standard error for this sample proportion. Remember the formula to estimate the standard error is ?SE?_est=v((p ^(1-p ^ ))/n).
Because the Central Limit Theorem applies, we know there is about a 95% chance that the sample proportion is within 2 standard errors of the actual population proportion. In other words, there is a 95% chance that the actual proportion of all American adults born between 1980 and 1983 is between .026-2SE and .026+2SE. Use your result from part c to determine this interval.
In Question 7, you calculated a confidence interval. Most of the time we will use StatCrunch to find confidence intervals.
Find a 95% confidence interval for the population proportion of all people considered having gifted natural intelligence. Round to the nearest tenth of a percent. (Hint: This answer should be very close to your answer in 7d!!)
Write a statement accurately interpreting the confidence interval.
Explain how the size of a confidence interval changes as the confidence level increases.
Illustrate your explanation in part c by finding an 80%, 90%, and 99% confidence interval for the population proportion of all adults considered having gifted natural intelligence. Round to the nearest tenth of a percent and use interval notation.
80% confidence interval
90% confidence interval
95% confidence interval
99% confidence interval
Suppose we define High Salary as an annual income of $75,000 or more.
What is the value of p ^, the sample proportion of adults with a high salary?
Find a 95% confidence interval for the population proportion of adults with a high salary.
Does this finding support a claim that 1 in 5 American adults born between 1980 and 1983 have a high salary?
Is there a difference in the percent of men and women deemed to have a high salary? To address this question:
Count the number of men and women with a high salary. Hint: See Question 8 from Activity 1 for a strategy to count using a frequency table.
Construct a 95% confidence interval for the difference in the proportion of men and women with a high salary. (Hint: Use Stat?Proportion Stats?Two Sample?With Summary. Success will be the counts you found in the frequency table from part a.)
Interpret the confidence interval.
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In this activity, you will once again use the IQ data file and StatCrunch to illustrate your understanding of a variety of concepts and skills learned in chapters 1-10. Before starting the activity, you should complete all assigned homework and quizzes through Chapter 10.
You will likely find this activity more challenging than the MyLab homework for several reasons. First off, you are working with an actual data set. You will need to spend a bit of time familiarizing yourself with the variables included in the set. Often students rely on the help aids or prompting included in algorithmically generated questions. None of those hints are present in the data set. Additionally, writing a sentence from scratch is more difficult, and more authentic, than selecting the correct word from a dropdown menu. Ask if you have questions. Complete every part of every question to the best of your ability. Expect to resubmit based upon instructor feedback.
I suggest you download this Word document and add your work directly to it. Click into the text box and add your answers to avoid Word renumbering the document. Provide an explanation of your work or thought process to arrive at your solution. Whenever you use StatCrunch, include screen captures of your output to support your answers. StatCrunch output alone is not sufficient – you must interpret the results. Once completed, save as a Word or pdf document and submit to the Dropbox in Talon.
Most questions in this activity require you to decide which test statistic is appropriate to compute for a given situation. As you have discovered in your work through the last several chapters, your test statistic choices are either a z-test statistic (Proportion Stats on StatCrunch) or t-test statistic (T Stats on StatCrunch). When asked, make sure you explain why you selected each particular test. Make sure to include an explanation of whether you have one or two samples, and if two samples, whether the samples are independent or dependent (paired).
About the data: Researchers want to investigate whether intelligence is associated with income and wealth. They sampled 500 American adults born between 1980 and 1983. The researchers administered an IQ test and computed IQ scores. A person is said to be naturally gifted if their IQ is 125 or above. Researchers also collected gender, 2018 annual income (rounded to nearest $500), and net worth at age 37 (rounded to nearest $10000).
1. Test the hypothesis that the proportion of all people considered having gifted natural intelligence is less than 4%. Use a .05 significance level.
a. State the null and alternative hypotheses.
b. Which test statistic is appropriate to compute for this situation? Why?
c. Compute the test statistic.
d. State and interpret the p-value for your hypothesis test.
2. Numerous sources state the average score on an IQ test is 100. Test the hypothesis that the mean IQ is not 100, using a significance level of .05.
a. State the null and alternative hypotheses.
b. Which test statistic is appropriate to compute for this situation? Why?
c. Compute the test statistic.
d. State and interpret the p-value for your hypothesis test.
3. Construct and interpret a 99% confidence interval for the mean IQ of American adults born between 1980 and 1983.
4. Construct and interpret a 95% confidence interval for the difference in mean annual income for men and women.
a. Construct the 95% confidence interval.
b. Interpret the confidence interval. Recall the most important aspect to consider is whether or not the interval includes 0.
5. Test the hypothesis that there is a difference in mean income for women and men. Use a significance level of .05.
a. Discuss any conditions that need to be satisfied for doing this investigation.
b. State the null and alternative hypotheses.
c. Which test statistic is appropriate to compute for this situation? Why?
d. Compute the test statistic.
e. State and interpret the p-value for your hypothesis test.
6. In Activity 2, question 9 we defined High Salary as an annual income of $75,000 or more. You determined 99 of 500 adults in the sample had a high salary. Suppose in 2021, researchers selected another random sample from the same population of American adults born between 1980 and 1983 and found 232 of 975 adults satisfied the High Salary definition. Test the hypothesis that the proportion of High Salary adults has increased. Use a significance level of .05.
a. Which test statistic is appropriate to compute for this situation? Why?
b. State the null and alternative hypotheses for this test.
c. Compute the test statistic.
d. State and interpret the p-value for your hypothesis test.