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Chapter 20: Problem 28

The U.S. Department of Commerce reported the results of a large-scale survey on high school graduation. Researchers contacted more than 25,000 Americans aged 24 years to see if they had finished high school; \(84.9 \%\) of the 12,460 males and \(88.1 \%\) of the 12,678 females indicated that they had high school diplomas. a. Are the assumptions and conditions necessary for inference satisfied? Explain. b. Create a \(95 \%\) confidence interval for the difference in graduation rates between males and females. c. Interpret your confidence interval. d. Does this provide strong evidence that girls are more likely than boys to complete high school? Explain.

Short Answer

Expert verified
Yes, the conditions are satisfied. The 95% confidence interval for the difference in graduation rates between males and females shows that females are more likely to graduate. This provides evidence that girls are more likely to complete high school, however other factors may come into play.

Step by step solution

01

Explain assumption and conditions

The assumptions and conditions for the inference include: 1. Randomization: We assume that the survey was conducted randomly. 2. Independence: We assume that the responses of the males and females are independent of each other.3. Normality: The sample size is greater than 10 which leads us to the assumption of normality for binomial distribution.4. Sample size: The sample size is large enough to apply the central limit theorem.
02

Calculate the difference in proportions

First, we calculate the proportions of males and females who have high school diplomas. For males: \(p1 = 0.849\) For females: \(p2 = 0.881\) The difference in proportions is \(p = p1 - p2 = 0.849 - 0.881 = -0.032\).
03

Calculate the standard error

The standard error (SE) of the difference in proportions is computed as \[\ SE = \sqrt{\frac{p1 \cdot (1 - p1)}{n1} + \frac{p2 \cdot (1 - p2)}{n2}}\] where \(n1 = 12460\) (number of males) and \(n2 = 12678\) (number of females). So, the calculation would be:\[\ SE = \sqrt{\frac{0.849 \cdot (1 - 0.849)}{12460} + \frac{0.881 \cdot (1 - 0.881)}{12678}}\]
04

Create a confidence interval

A 95% confidence interval for the difference in proportions is given as \[p \pm Z \cdot SE\] where Z is the z-score corresponding to a 95% confidence level which is 1.96. Substituting the values we have, the interval would be calculated as:\[-0.032 \pm 1.96 \cdot SE\]
05

Interpret the confidence interval

The 95% confidence interval indicates that we are 95% confident that the difference in graduation rates between males and females lies within this interval. A negative value means that the graduation rate for females is higher.
06

Conclude our findings

Since the confidence interval does not include 0, this provides strong evidence that girls are more likely than boys to complete high school. However, it's always important to remember that correlation does not imply causation. This data only shows a relationship and does not account for other factors that may have an impact on high school completion.

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Key Concepts

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

High School Graduation Rates
High school graduation is a critical milestone in a student's educational journey. It provides them the necessary foundation for further education or entry into the workforce. In the U.S., organizations gather data to analyze graduation rates to understand educational trends and challenges.

In this exercise, the high school graduation rates among males and females are surveyed. The reported rates were 84.9% for males and 88.1% for females. This data can be further analyzed to determine whether there is a significant difference between the graduation rates of the two groups.

Understanding graduation rates can aid in assessing educational policies and identifying areas needing improvement. It also helps to ensure equitable access to education for all demographics.
Difference in Proportions
The concept of 'difference in proportions' allows us to quantify the difference between two groups' outcomes. In our case, we look at the graduation rates of males versus females. The proportion for males is 0.849, and for females, it is 0.881.

To find the difference in proportions, we subtract the male graduation rate from the female graduation rate: \[p = 0.849 - 0.881 = -0.032\]

A negative difference indicates that the proportion (or rate) for females is higher than that for males. This calculation helps us measure how much more likely, on average, females are to graduate compared to males.
Statistical Inference
Statistical inference is the process of using data from a sample to make conclusions about a larger population. It involves several key assumptions to ensure valid results:
  • Randomization: The data should be collected randomly to avoid bias.
  • Independence: Each observation must be independent. In our scenario, the male and female responses are assumed independent.
  • Normality: For proportions, the sample size should be large enough that the distribution approximates normality.
Applying statistical inference allows researchers to draw conclusions about high school graduation rates based on the sample from the survey. It helps in understanding if the observed difference is significant or just due to random chance.
Central Limit Theorem
The Central Limit Theorem (CLT) is a fundamental concept in statistics. It states that, given a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the original distribution.

In our analysis, the sample sizes (over 12,000 for each group) are large enough to apply the CLT. This means that the distribution of graduation rates can be treated as normal, facilitating the computation of the standard error and the creation of a confidence interval.

The CLT ensures that even if the original population distribution is unknown, we can still carry out statistical analyses and make inferences about the graduation rates with confidence. This principle underpins many of the procedures used in hypothesis testing and confidence interval estimation.

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Most popular questions from this chapter

A consumer magazine plans to poll car owners to see if they are happy enough with their vehicles that they would purchase the same model again. They'll randomly select 450 owners of American-made cars and 450 owners of Japanese models. Obviously, the actual opinions of the entire population couldn't be known, but suppose \(76 \%\) of owners of American cars and \(78 \%\) of owners of Japanese cars would purchase another. a. What kind of sampling design is the magazine planning to use? b. What difference would you expect their poll to show? c. Of course, sampling error means the poll won't reflect the difference perfectly. What's the standard deviation for the difference in the proportions? d. Sketch a sampling model for the difference in proportions that might appear in a poll like this. e. Could the magazine be misled by the poll, concluding that owners of American cars are much happier with their vehicles than owners of Japanese cars? Explain.

In Chapter 6 , Exercise 25 , we looked at collected samples of water from streams in the Adirondack Mountains to investigate the effects of acid rain. Researchers measured the pH (acidity) of the water and classified the streams with respect to the kind of substrate (type of rock over which they flow). A lower pH means the water is more acidic. Here is a boxplot of the \(\mathrm{pH}\) of the streams by substrate (limestone, mixed, or shale): Here are selected parts of a software analysis comparing the pH of streams with limestone and shale substrates: 2 -Sample \(t\) -Test of \(\mu_{1}-\mu_{2}\) Difference Between Means \(=0.735\) \(t\) -Statistic \(=16.30 \mathrm{w} / 133 \mathrm{df}\) \(\mathrm{p} \leq 0.0001\) a. State the null and alternative hypotheses for this test. b. From the information you have, do the assumptions and conditions appear to be met? c. What conclusion would you draw?

Data collected in 2015 by the Behavioral Risk Factor Surveillance System revealed that in the state of New Jersey, \(27.3 \%\) of whites and \(47.2 \%\) of blacks were cigarette smokers. Suppose these proportions were based on samples of 3607 whites and 485 blacks. a. Create a \(90 \%\) confidence interval for the difference in the percentage of smokers between black and white adults in New Jersey. b. Does this survey indicate a race-based difference in smoking among American adults? Explain, using your confidence interval to test an appropriate hypothesis. c. What alpha level did your test use?

Political pundits talk about the "bounce" that a presidential candidate gets after his party's convention. In the past 40 years, it has averaged about 6 percentage points. Just before the 2004 Democratic convention, Rasmussen Reports polled 1500 likely voters at random and found that \(47 \%\) favored John Kerry. Just afterward, they took another random sample of 1500 likely voters and found that \(49 \%\) favored Kerry. That's a two percentage point increase, but the pollsters claimed that there was no bounce. Explain.

When a random sample of 935 parents were asked about rules in their homes, \(77 \%\) said they had rules about the kinds of TV shows their children could watch. Among the 790 of those parents whose teenage children had Internet access, \(85 \%\) had rules about the kinds of Internet sites their teens could visit. That looks like a difference, but can we tell? Explain why a two-sample \(z\) -test would not be appropriate here.
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