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Chapter 9: Problem 4

What requirements must be satisfied in order to construct a confidence interval about a population mean?

Short Answer

Expert verified
Large sample size, random sampling, normality for small samples, knowledge of standard deviation, and independence of observations.

Step by step solution

01

- Large Sample Size

Ensure that the sample size is large enough, typically n 鈮� 30, to use the Central Limit Theorem. This helps ensure that the sampling distribution of the sample mean will be approximately normally distributed, which is necessary for constructing a confidence interval.
02

- Random Sampling

The sample should be randomly selected from the population. Random sampling ensures that the sample is representative of the population, which is crucial for the validity of the confidence interval.
03

- Normality of Population Distribution

If the sample size is small (n < 30), the population from which the sample is drawn should be normally distributed. This allows the use of T-distribution for constructing the confidence interval.
04

- Known or Unknown Population Standard Deviation

Determine whether the population standard deviation (蟽) is known. If 蟽 is known, the Z-distribution is used. If 蟽 is unknown and n 鈮� 30, the sample standard deviation (s) is used in place of 蟽, and the T-distribution is used.
05

- Independence of Observations

Ensure that the observations in the sample are independent of each other. This means that the selection of one individual does not influence the selection of another.

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Key Concepts

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

Large Sample Size
To construct a reliable confidence interval, having a large sample size is crucial. A good rule of thumb is to have at least 30 observations ( 鈮� 30).
This is because the Central Limit Theorem tells us that for a sufficiently large sample size, the distribution of the sample mean will approximate a normal distribution, regardless of the population's distribution.
This normality assumption simplifies calculations and ensures the accuracy of the confidence interval.
Random Sampling
Random sampling is essential to ensure that your sample accurately represents the population.
This means that every member of the population has an equal chance of being selected.
Without random sampling, your confidence interval may not be valid, as it could be biased.
  • It helps in generalizing the results to the entire population.
  • Reduces sampling bias and increases the precision of the confidence interval.
Normality of Population Distribution
If your sample size is less than 30, the normality of the population becomes important.
For smaller samples, we can't rely on the Central Limit Theorem, so we need the population to be normally distributed.
If the population is normal, even a small sample will give accurate results. For these cases, we use the T-distribution instead of the Z-distribution for constructing confidence intervals.
Remember: < 30 -> Check for normality, 鈮� 30 -> Normality check less critical.
Known or Unknown Population Standard Deviation
Whether you know the population standard deviation () affects how to calculate the confidence interval.
If is known, use the Z-distribution.
If is unknown and the sample size is large ( 鈮� 30), the sample standard deviation (s) is used with the T-distribution.
This substitution allows for more flexibility in real-world scenarios, where is often unknown.
  • Known -> Z-distribution
  • Unknown , 鈮� 30 -> T-distribution
Independence of Observations
Independence of observations is a critical assumption when constructing a confidence interval.
This means that the selection of one observation should not influence the selection of another.
If this assumption is violated, the results can be misleading. Ensuring independence can often be achieved through proper sampling methods, such as random sampling.
It's a safeguard to ensure the validity of the confidence interval.
  • Helps in reducing biases.
  • Increases the reliability of the conclusions.

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Most popular questions from this chapter

In a random sample of 40 felons convicted of aggravated assault, it was determined that the mean length of sentencing was 54 months, with a standard deviation of 8 months. Construct and interpret a \(95 \%\) confidence interval for the mean length of sentence for an aggravated assault conviction.

Construct the appropriate confidence interval. A simple random sample of size \(n=785\) adults was asked if they follow college football. Of the 785 surveyed, 275 responded that they did follow college football. Construct a \(95 \%\) confidence interval for the population proportion of adults who follow college football.

True or False: To construct a confidence interval about the mean, the population from which the sample is drawn must be approximately normal.

Blood Alcohol Concentration A random sample of 51 fatal crashes in 2013 in which the driver had a positive blood alcohol concentration (BAC) from the National Highway Traffic Safety Administration results in a mean BAC of 0.167 gram per deciliter \((\mathrm{g} / \mathrm{dL})\) with a standard deviation of \(0.010 \mathrm{~g} / \mathrm{dL}\) (a) A histogram of blood alcohol concentrations in fatal accidents shows that BACs are highly skewed right. Explain why a large sample size is needed to construct a confidence interval for the mean BAC of fatal crashes with a positive \(\mathrm{BAC}\) (b) In \(2013,\) there were approximately 25,000 fatal crashes in which the driver had a positive BAC. Explain why this, along with the fact that the data were obtained using a simple random sample, satisfies the requirements for constructing a confidence interval. (c) Determine and interpret a \(90 \%\) confidence interval for the mean BAC in fatal crashes in which the driver had a positive BAC. (d) All 50 states and the District of Columbia use a BAC of \(0.08 \mathrm{~g} / \mathrm{dL}\) as the legal intoxication level. Is it possible that the mean BAC of all drivers involved in fatal accidents who are found to have positive BAC values is less than the legal intoxication level? Explain.

The Sullivan Statistics Survey II asks, "What percent of one's income should an individual pay in federal income taxes?" Go to www.pearsonhighered.com/sullivanstats to obtain the data file SullivanStatsSurveyII using the file format of your choice for the version of the text you are using. The data is in the column "Tax Rate" (a) Draw a relative frequency histogram of the variable "Tax Rate" using a lower class limit of the first class of 0 and a class width of \(5 .\) Comment on the shape of the distribution. (b) Draw a boxplot of the variable "Tax Rate." Are there any outliers? (c) Explain why a large sample is necessary to construct a confidence interval for the mean tax rate. (d) Treat the respondents of this survey as a simple random sample of U.S. residents. Use statistical software to construct and interpret a \(90 \%\) confidence interval for the mean tax rate U.S. residents feel an individual should pay in federal income taxes.
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