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Chapter 11: Problem 32

In June \(2014,\) the Gallup organization surveyed 1134 American adults and found that 298 had confidence in public schools. In June \(2013,\) the Gallup organization had surveyed 1134 American adults and found that 360 had confidence in public schools. Suppose that a newspaper article has a headline that reads, "Confidence in Public Schools Deteriorates." Is this an accurate headline? Why?

Short Answer

Expert verified
Yes, the headline is accurate as the confidence dropped from 31.7% in 2013 to 26.3% in 2014.

Step by step solution

01

- Calculate the Proportion for 2013

The proportion of American adults who had confidence in public schools in 2013 can be calculated by dividing the number of adults with confidence by the total number of surveyed adults. For 2013, the calculation is: \[\text{Proportion}_{2013} = \frac{360}{1134}\] Solving this gives: \[\text{Proportion}_{2013} = 0.317\]
02

- Calculate the Proportion for 2014

Similarly, the proportion of American adults who had confidence in public schools in 2014 can be calculated by dividing the number of adults with confidence by the total number of surveyed adults. For 2014, the calculation is: \[\text{Proportion}_{2014} = \frac{298}{1134}\] Solving this gives: \[\text{Proportion}_{2014} = 0.263\]
03

- Compare the Proportions

To determine if the confidence in public schools has deteriorated, compare the proportions from 2013 and 2014: \[0.317 - 0.263 = 0.054\] This shows a decrease in the proportion of American adults with confidence in public schools from 31.7% in 2013 to 26.3% in 2014.
04

- Interpret the Results

Since there is a decrease in the proportion of American adults who have confidence in public schools from 2013 to 2014, it suggests that the confidence has indeed deteriorated. The headline 'Confidence in Public Schools Deteriorates' accurately reflects this change.

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Key Concepts

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

Confidence Intervals
Confidence intervals are a range of values used to estimate a population parameter. They provide an interval within which we can be certain, to a specified probability, that the true parameter lies. For instance, in our Gallup poll example, instead of just stating that 26.3% of respondents had confidence in public schools in 2014, we might want to know if our sample proportion accurately reflects the entire population鈥檚 confidence.

By using a confidence interval, we can determine if the 2014 confidence rate is significantly different from the 2013 rate. This interval helps us to see the margin of error and adds more credibility to our comparison. Understanding this concept is essential for interpreting survey results accurately.
Survey Data Analysis
Survey data analysis involves collecting responses from a sample group and deriving insights from this data. In the given exercise, the Gallup organization surveyed 1134 American adults each year. By comparing the proportions obtained from these surveys, analysts can infer trends and shifts in public opinion.

An important aspect of survey data analysis is ensuring that the sample represents the population. This involves looking into the size and method of sampling. Furthermore, analyzing data includes calculating proportions, means, and other statistics to understand the bigger picture. The goal is to draw conclusions that reflect the population and inform decisions or opinions, just as the Gallup poll did for public confidence in schools.
Proportion Comparison
Proportion comparison is a method used to determine whether there is a significant difference between two or more proportions. In our exercise, we calculated the proportion of adults who had confidence in public schools for 2013 and 2014. By comparing these proportions鈥�31.7% in 2013 and 26.3% in 2014鈥攚e observed a noticeable drop of 5.4%.

However, to determine if this difference is statistically significant, more advanced methods such as hypothesis testing or calculating the confidence interval around the difference would be used. These steps ensure that the observed difference isn't just due to random fluctuations or sample variability.

Understanding how to compare proportions effectively is essential for correctly interpreting changes and trends in survey data.
Gallup Poll
Gallup polls are well-known surveys conducted to gauge public opinion on various topics. These polls employ rigorous sampling and data collection methodologies to ensure the reliability and validity of the results. In this exercise, the Gallup organization surveyed exact numbers (1134) of American adults each year, emphasizing its methodical approach.

Gallup polls help in capturing snapshot views of the population at different times, which can show trends, as seen with the decrease in confidence in public schools from 2013 to 2014. They often influence public discourse, policy-making and give insights to stakeholders about public sentiment.

Understanding Gallup polls and their processes is important for anyone looking to grasp public opinion with accuracy and accuracy and reliability.

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Most popular questions from this chapter

In Problems 3鈥�8, determine whether the sampling is dependent or independent. Indicate whether the response variable is qualitative or quantitative. A sociologist wishes to compare the annual salaries of married couples in which both spouses work and determines each spouse鈥檚 annual salary.

Government Waste In a Gallup poll 513 national adults aged 18 years or older who consider themselves to be Republican were asked, "Of every tax dollar that goes to the federal government in Washington, D.C., how many cents of each dollar would you say are wasted?" The mean wasted was found to be 54 cents with a standard deviation of 2.9 cents. The same question was asked of 513 national adults aged 18 years or older who consider themselves to be Democrat. The mean wasted was found to be 41 cents with a standard deviation of 2.6 cents. Construct a \(95 \%\) confidence interval for the mean difference in government waste, \(\mu_{R}-\mu_{D} .\) Interpret the interval.

Sugary Beverages It has been reported that consumption of sodas and other sugar-sweetened beverages cause excessive weight gain. Researchers conducted a randomized study in which 224 overweight and obese adolescents who regularly consumed sugar-sweetened beverages were randomly assigned to experimental and control groups. The experimental groups received a one-year intervention designed to decrease consumption of sugar-sweetened beverages, with follow-up for an additional year without intervention. The response variable in the study was body mass index (BMI-the weight in kilograms divided by the square of the height in meters). Results of the study appear in the following table. Source: Cara B. Ebbeling, \(\mathrm{PhD}\) and associates, "A Randomized Trial of Sugar-Sweetened Beverages and Adolescent Body Weight" N Engl J Med 2012;367:1407-16. DOI: 10.1056/NEJMoal2Q3388 $$ \begin{array}{lll} & \text { Experimental } & \text { Control } \\ & \text { Group } & \text { Group } \\ & (n=110) & (n=114) \\ \hline \text { Start of Study } & \text { Mean BMI }=30.4 & \text { Mean BMI }=30.1 \\ & \text { Standard Deviation } & \text { Standard Deviation } \\ & \mathrm{BMI}=5.2 & \mathrm{BMI}=4.7 \\ \hline \text { After One Year } & \text { Mean Change in } & \text { Mean Change in } \\ & \mathrm{BMI}=0.06 & \mathrm{BMI}=0.63 \\ & \text { Standard Deviation } & \text { Standard Deviation } \\ & \text { Change in } & \text { Change in } \\ & \mathrm{BMI}=0.20 & \mathrm{BMI}=0.20 \\ \hline \text { After Two Years } & \text { Mean Change in } & \text { Mean Change in } \\ & \mathrm{BMI}=0.71 & \mathrm{BMI}=1.00 \\ & \text { Standard Deviation } & \text { Standard Deviation } \\ & \text { Change in } & \text { Change in } \\ & \mathrm{BMI}=0.28 & \mathrm{BMI}=0.28 \end{array} $$ (a) What type of experimental design is this? (b) What is the response variable? What is the explanatory variable? (c) One aspect of statistical studies is to verify that the subjects in the various treatment groups are similar. Does the sample evidence support the belief that the BMIs of the subjects in the experimental group is not different from the BMIs in the control group at the start of the study? Use an \(\alpha=0.05\) level of significance. (d) One goal of the research was to determine if the change in BMI for the experimental group was less than that for the control group after one year. Conduct the appropriate test to see if the evidence suggests this goal was met. Use an \(\alpha=0.05\) level of significance. What does this result suggest? (e) Does the sample evidence suggest the change in BMI is less for the experimental group than the control group after two years? Use an \(\alpha=0.05\) level of significance. What does this result suggest? (f) To what population do the results of this study apply?

For each study, explain which statistical procedure (estimating a single proportion; estimating a single mean; hypothesis test for a single proportion; hypothesis test for a single mean; hypothesis test or estimation of two proportions, hypothesis test or estimation of two means, dependent or independent) would most likely be used for the research objective given. Assume all model requirements for conducting the appropriate procedure have been satisfied. Does Marriott Courtyard charge more than Holiday Inn Express for a one-night stay?

An experiment was conducted in which 16 ten-month-old babies were asked to watch a climber character attempt to ascend a hill. On two occasions, the baby witnesses the character fail to make the climb. On the third attempt, the baby witnesses either a helper toy push the character up the hill, or a hinderer toy preventing the character from making the ascent. The helper and hinderer toys were shown to each baby in a random fashion for a fixed amount of time. In Problem 39 from Section \(10.2,\) we learned that, after watching both the helper and hinderer toy in action, 14 of 16 ten-month-old babies preferred to play with the helper toy when given a choice as to which toy to play with. A second part of this experiment showed the climber approach the helper toy, which is not a surprising action, and then alternatively the climber approached the hinderer toy, which is a surprising action. The amount of time the ten-month-old watched the event was recorded. The mean difference in time spent watching the climber approach the hinderer toy versus watching the climber approach the helper toy was 1.14 seconds with a standard deviation of 1.75 second. Source: J. Kiley Hamlin et al. "Social Evaluation by Preverbal Infants," Nature, Nov. \(2007 .\) (a) State the null and alternative hypothesis to determine if babies tend to look at the hinderer toy longer than the helper toy. (b) Assuming the differences are normally distributed with no outliers, test if the difference in the amount of time the baby will watch the hinderer toy versus the helper toy is greater than 0 at the 0.05 level of significance. (c) What do you think the results of this experiment imply about 10 -month- olds' ability to assess surprising behavior?
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