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Chapter 14: Problem 474

State briefly the assumptions involved in establishing a confidence interval using the standard normal distribution and the \(t\) distribution.

Short Answer

Expert verified
To establish a confidence interval using the standard normal distribution, we assume that: (1) the population is normally distributed, (2) the population standard deviation is known, and (3) the sample is a simple random sample. For the t distribution, we assume that: (1) the population is normally distributed, (2) the population standard deviation is unknown, (3) the sample is a simple random sample, and (4) the sample size is relatively small (usually n < 30). The key difference between the two distributions is whether the population standard deviation is known or not.

Step by step solution

01

Assumptions for Standard Normal Distribution Confidence Interval

To establish a confidence interval using the standard normal distribution, we need to make the following assumptions: 1. The population from which the sample is taken is normally distributed. This is important because the standard normal distribution's properties only apply to normally distributed data. 2. The population standard deviation, 蟽, is known. In a standard normal distribution, we use the known population standard deviation to calculate the standard error and create the confidence interval. 3. The sample is a simple random sample. This means that every individual in the population has an equal probability of being selected for the sample, ensuring independence between data points.
02

Assumptions for t Distribution Confidence Interval

To establish a confidence interval using the t distribution, we need to make the following assumptions: 1. The population from which the sample is taken follows a normal distribution. Similar to the standard normal distribution, it is essential that the population follows a normal distribution to apply the properties of the t distribution. 2. The population standard deviation, 蟽, is unknown. In contrast to the standard normal distribution, we use the t distribution when we don't know the population standard deviation. Instead, we rely on the sample standard deviation, s, as an estimate of 蟽. 3. The sample is a simple random sample to ensure independence between data points. 4. Sample size n is relatively small (usually n < 30). When sample size is large, both distributions become very similar as the t distribution approaches the standard normal distribution, and we can use the standard normal distribution instead. In conclusion, establishing a confidence interval using the standard normal distribution and the t distribution involves several assumptions related to the properties of the population and the data sampling methods. The key distinction between the two distributions is whether the population standard deviation is known. If it is known, we can use the standard normal distribution. If it is unknown and the sample size is small, we need to use the t distribution.

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Key Concepts

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

Standard Normal Distribution
Understanding the standard normal distribution is fundamental for statistics students, especially when it comes to creating confidence intervals. Imagine a bell-shaped curve where most of the data clusters around the mean, and this curve is symmetrical. This is a graphical representation of the normal distribution. The standard normal distribution is a special case where the mean is 0, and the standard deviation is 1.

When your data follows this pattern, you can use the standard normal distribution to calculate probabilities and confidence intervals. Here's the catch, though: you need to know the population standard deviation (蟽). This allows you to standardize your sample data and use z-scores to find the confidence interval. A z-score tells you how many standard deviations a data point is from the mean. If the standard normal distribution assumptions hold, these intervals can provide highly reliable estimates for the population parameters.

Remembering the Essential Conditions

  • The population must follow a normal distribution.
  • The population standard deviation is known.
  • Sampling should be random to avoid bias.
Making sure these conditions are met is like checking if you have the right key for a lock鈥攚ithout them, you can't unlock the true insights from your data.
T Distribution
Now, let's talk about the t distribution, a close relative of the standard normal distribution, but it's typically used when the sample size is small and/or the population standard deviation is unknown. It's like a safety net for when you don't have all the information about your population.

The t distribution has heavier tails than the standard normal distribution. This means that it allows for a greater possibility of extreme values in your data, which is common when you're dealing with small sample sizes or uncertain population parameters. As the sample size grows, the t distribution starts to look more like the standard normal distribution, and the two become almost indistinguishable with larger samples.

Meeting the Requirements

  • Assumes population is normally distributed.
  • Used when the population standard deviation is unknown, with sample standard deviation as a stand-in.
  • Random sampling is just as crucial here.
  • Typically employed when sample sizes are smaller (usually n < 30).
One way to remember when to use the t distribution is that it has an extra 'T' for 'Tiny samples' or 'Total mystery' when it comes to the population standard deviation.
Population Standard Deviation
Finally, the population standard deviation (denoted as 蟽) is a measure that tells us how spread out the values in a population are from the mean. In more accessible terms, it shows us how much wiggle room there is in the data. High standard deviation means your data points are spread out; a low standard deviation means they are clumped together around the mean.

Knowing the population standard deviation is ideal because it increases the accuracy of your confidence intervals. But in real-world situations, it's a rare luxury. When you don't know 蟽, you must rely on your sample data to provide an estimate. This is when the standard deviation of your sample (denoted as s) steps in to give you a rough idea of 蟽.

Extra Insights

  • Population standard deviation is the square root of the variance of the population.
  • In practice, 蟽 is often unknown, making sample standard deviation a vital substitute.
  • The accuracy of s as an estimator for 蟽 increases with larger sample sizes.
Knowing whether you have the population standard deviation or not dictates whether you'll be working with z-scores or t-scores in your statistical analysis and interpreting the data based on the most appropriate distribution.

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Most popular questions from this chapter

In Grand Central Station, there is a coffee machine which is regulated so that the amount of coffee dispensed is normally distributed with a standard deviation of \(.5\) ounces per cup. A random sample of 50 cups had an average of 5 ounces per cup. (a) Find the maximum likelihood estimate for the average amount of coffee in each cup dispensed by the machine. (b) Find a \(95 \%\) confidence interval for the mean of all cups dispensed.

A random sample of 50 workers is taken out of a very large number of workers in a factory; the time that each of the workers in the sample takes to perform the same manufacturing process is recorded. The average time requirement for this sample is 21 minutes and the standard deviation is 3 minutes. Find the \(99 \%\) confidence interval for the average time requirement to perform this manufacturing process for all the workers in this factory.

A standardized chemistry test was given to 50 girls and 75 boys. The girls made an average grade of 76 with a standard deviation of 6, while the boys made an average grade of 82 with a standard deviation of \(8 .\) Find a \(96 \%\) confidence interval for the difference \(\mu_{\mathrm{X}}-\mu_{\mathrm{Y}}\) where \(\mu_{\mathrm{X}}\) is the mean score of all boys and \(\mu_{\mathrm{Y}}\) is the mean score of all girls who might take this test.

The Harvard class of 1927 had a reunion which 36 attended. Among them they discovered they had been married an average of \(2.6\) times apiece. From the Harvard Alumni Register Dean Epps learned that the standard deviation for the 1927 alumni was \(0.3\) marriages. Help Dean Epps construct a 99 per cent confidence interval for the marriage rate of all Harvard alumni.

A random sample of 100 students from a large college showed an average IQ score of 112 with a standard deviation of 10 . (a) Establish a 95 confidence interval estimate of the mean IQ score of all students attending this college. (b) Establish a \(.99\) confidence interval estimate of the mean IQ score of all students in this college.
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