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Chapter 9: Problem 52

A \(95 \%\) confidence interval for the global concentration of carbon monoxide is \(\left(0-0.06 \mathrm{mg} / \mathrm{m}^{3}\right)\). Either interpret the interval or explain why it should not be interpreted.

Short Answer

Expert verified
This confidence interval means that we're 95% confident that the true average global concentration of carbon monoxide falls between 0 and 0.06 mg/m^3. However, this doesn't mean that 95% of the time the concentration lies within this range.

Step by step solution

01

Understanding Confidence Interval

A confidence interval provides a range of values which is likely to contain the population parameter. The confidence level of 95% tells us that if we were to repeat the sampling process, approximately 95 out of 100 produced confidence intervals would contain the true population parameter.
02

Interpreting the Confidence Interval

This given confidence interval can be observed as: The concentration of carbon monoxide is between 0 and 0.06 mg/m^3, with a 95% confidence. This means there is a 95% chance that this interval contains the true average global concentration of carbon monoxide.
03

Though about Extreme Ends

However, keep in mind it doesn't mean that 95% of the time, the concentration of the carbon monoxide in the globe lies within this range.

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Key Concepts

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

Population Parameter
In statistics, the term "population parameter" refers to a numeric value that represents a characteristic of an entire population. In the case of carbon monoxide concentration, the population parameter would be the average concentration level across the entire globe.

While it is impossible to measure this exact value without covering every square meter of Earth's surface, statistical methods allow us to approximate this parameter. We usually estimate it using sample data. This estimated range gives us an idea of where the true parameter likely lies, emphasizing the importance of accurate data collection.
  • The population parameter is the actual value we're trying to understand through our study or analysis.
  • It remains constant unless the population changes significantly or the parameter we're studying changes over time.
  • Confidence intervals are statistical tools used to guess where the parameter might fall, based on sample data.
Carbon Monoxide Concentration
Carbon monoxide (CO) is a colorless and odorless gas that can have significant effects on both human health and the environment. It is produced largely by the burning of fossil fuels and is a concern for air quality standards worldwide.

Determining the concentration of CO involves measuring how much of this gas is present in a specific volume of air, usually indicated in mg/m鲁 (milligrams per cubic meter). Understanding its concentration globally is crucial, as this gas can contribute to issues like urban smog and poses health hazards when concentrations are too high.
  • Global average concentrations provide insight into pollution levels and help create strategies for environmental protection.
  • Monitoring CO levels is a key aspect of evaluating air quality standards and the effectiveness of pollution control policies.
  • Accurate estimation of CO concentration involves various scientific methods and statistical models, often involving data from different parts of the world.
Statistical Inference
Statistical inference involves using data from a sample to make educated guesses or conclusions about a broader population. In the context of estimating global carbon monoxide concentration, statistical inference helps make predictions about the true average concentration using sampled measurements.

One of the main tools of statistical inference is the confidence interval. This provides a range of values which are believed to contain the population parameter, like the true concentration of carbon monoxide, with a specified level of confidence - such as 95% in typical scenarios.
  • Statistical inference bridges the gap between sample data and population knowledge through mathematical models and sampling techniques.
  • It not only helps estimate population parameters but also evaluates the reliability of these estimates by giving them confidence levels.
  • Understanding statistical inference is crucial for correctly interpreting data, such as distinguishing between mere variability in samples and actual trends in the population.

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Most popular questions from this chapter

Four-year Graduation Rate (Example 6) A random sample of 10 colleges from Kiplinger's 100 Best Values in Public Education was taken. The mean rate of graduation within four years was \(43.5 \%\) with a margin of error of \(6.0 \%\). The distribution of graduation rates is Normal. (Source: http://portal.kiplinger.com/tool/ college/T014-S001-kiplinger-s-best-values- in-public-colleges/index php#colleges. Accessed via StatCrunch. Owner: Webster West.) a. Decide whether each of the following statements is worded correctly for the confidence interval, and fill in the blanks for the correctly worded one(s). i. We are \(95 \%\) confident that the sample mean is between \(\%\) and \(\%\). ii. We are \(95 \%\) confident that the population mean is between \(-\%\) and \(\%\). iii. There is a \(95 \%\) probability that the population mean is between \(\%\) and \(\% .\) b. Can we reject a population mean percentage of \(50 \%\) on the basis of these numbers? Explain.

Hamburgers (Example 9) A hamburger chain sells large hamburgers. When we take a sample of 30 hamburgers and weigh them, we find that the mean is \(0.51\) pounds and the standard deviation is \(0.2\) pound. a. State how you would fill in the numbers below to do the calculation. with Minitab. b. Report the confidence interval in a carefully worded sentence. Normal.

Number of Children A random sample of 100 women from the General Social Survey showed that the mean number of children reported was \(1.85\) with a standard deviation of 1.5. (Interestingly, a sample of 100 men showed a mean of 1\. 49 children.) a. Find a \(95 \%\) confidence interval for the population mean number of children for women. Because the sample size is so large, you can use \(1.96\) for the critical value of \(t\) (which is the same as the critical value of \(z\) ) if you do the calculations manually. b. Find a \(90 \%\) confidence interval. Use \(1.645\) for the critical value of \(t\), which is the critical value of \(z\) c. Which interval is wider, and why?

Sale of Microwaves The average number of microwaves sold in 2015 was 2700 per day in the same city, and that was larger than the average for any other appliance but less than that of the air conditioners. Suppose the standard deviation is 1551 and the distribution is right-skewed. Suppose we take a random sample of 121 days in the year. a. What value should we expect for the sample mean? Why? b. What is the standard error for the sample mean?

Independent or Paired (Example 13) State whether each situation has independent or paired (dependent) samples. a. A researcher wants to know whether pulse rates of people go down after brief meditation. She collects the pulse rates of a random sample of people before meditation and then collects their pulse rates after meditation. b. A researcher wants to know whether women who text send more text messages than men who text. She gathers two random samples, one from men and one from women, and asks them how many text messages they sent yesterday.
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