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Chapter 6: Problem 72

Use a t-distribution to answer the question. Assume the sample is a random sample from a distribution that is reason ably normally distributed and we are doing inference for a sample mean. Find the area in a t-distribution above 1.5 if the sample has size \(n=8\).

Short Answer

Expert verified
The area in a t-distribution above 1.5 for a sample size of 8 can be found using a t-distribution table or calculator. Since the exact value may vary based on the table or calculator you are using, it will roughly be the value obtained from \(1 - P(T \leq 1.5)\) where \(P(T \leq 1.5)\) is the probability you came up with from the t-distribution table or calculator.

Step by step solution

01

Identify Degrees of Freedom

The degrees of freedom for a t-distribution is identified by the following formula: \(df = n - 1\). Here, the sample size is 8. Therefore, the degrees of freedom will be \(df = 8 - 1 = 7\). This value is important for referring to the correct row in the t-distribution table.
02

Refer to Correct row in t-distribution table

Having identified the degrees of freedom as 7, the next step is to add this to the t-distribution table and locate the correct row. After locating the correct row, move to the right until reaching the t-score of 1.5.
03

Determine the Area in the t-Distribution

After locating the t-score of 1.5 in the row for df=7, the corresponding probability under the t-distribution curve up to that point is typically recorded in the table or calculator. However, we want the area 'above' not 'below' this t-score. As the entire area under the curve should sum to 1, you will find the desire area by subtracting the identified probability from 1.

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Key Concepts

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

Degrees of Freedom
Degrees of freedom is a concept that helps us determine which version of a statistical distribution we should use when analyzing data. In the context of a t-distribution, degrees of freedom are calculated using the formula: \( df = n - 1 \). Here, \(n\) is the size of your sample. For example, if you have a sample size of 8, the degrees of freedom is \(8 - 1 = 7\). This means you'll refer to the specific version of the t-distribution table that accounts for these 7 degrees of freedom.
  • Degrees of freedom help tailor the t-distribution to better fit smaller sample sizes.
  • Understanding degrees of freedom is crucial to analyzing sample data correctly.
  • Accurately finding your degrees of freedom ensures you use the right statistical table values.
Each t-distribution is slightly different based on the degrees of freedom, making it essential to pinpoint the correct one for reliable analysis.
Inference for a Sample Mean
Inference for a sample mean involves using statistics to make inferences about a population mean based on a sample mean. When our sample size is relatively small and the population standard deviation is unknown, the t-distribution is particularly useful. It allows us to estimate probabilities and confidently make inferences.
  • The process involves calculating a sample mean and using that to estimate the population mean.
  • We utilize the t-distribution because it adjusts for small samples, offering a more accurate reflection of variability compared to the normal distribution.
  • This method includes constructing confidence intervals or conducting hypothesis tests related to the population mean.
Inference for a sample mean is a foundational idea in statistics and vital for making informed conclusions about larger populations from smaller groups.
Probability
Probability in the context of a t-distribution refers to the chance of observing a t-value within a certain range. When you have a t-score, like 1.5, the task often involves finding the area under this score. This area represents the cumulative probability up to that t-score.
  • It's essential to differentiate between the probability 'above' and 'below' a t-score.
  • The t-distribution table typically provides the probability up to the t-score.
  • To find the area above, subtract the given probability from 1.
Understanding probability helps in interpreting results correctly and making meaningful inferences based on data analysis. Being able to calculate these probabilities is vital for hypothesis testing and other statistical analyses.

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Most popular questions from this chapter

Football Air Pressure During the National Football League's 2014 AFC championship game, officials measured the air pressure on 11 of the game footballs being used by the New England Patriots. They found that the balls had an average air pressure of 11.1 psi, with a standard deviation of 0.40 psi. (a) Assuming this is a representative sample of all footballs used by the Patriots in the 2014 season, perform the appropriate test to determine if the average air pressure in footballs used by the Patriots was significantly less than the allowable limit of 12.5 psi. There is no extreme skewness or outliers in the data, so it is appropriate to use the \(\mathrm{t}\) -distribution. (b) Is it fair to assume that this sample is representative of all footballs used by the Patriots during the 2014 season?

Use a t-distribution to answer the question. Assume the sample is a random sample from a distribution that is reason ably normally distributed and we are doing inference for a sample mean. Find the area in a t-distribution above 2.3 if the sample has size \(n=6\).

6.122 Be Nice to Pigeons, As They Remember Your Face In a study \(^{30}\) conducted in Paris, France, equal amounts of pigeon feed were spread on the ground in two adjacent locations. A person was present in both sites, with one acting hostile and running at the birds to scare them away and the other acting neutral and just observing. The two people were randomly exchanged between the two sites throughout and the birds quickly learned to avoid the hostile person's site and to eat at the site of the neutral person. At the end of the training session, both people behaved neutrally but the birds continued to remember which one was hostile. In the most interesting part of the experiment, when the two people exchanged coats (orange worn by the hostile one and yellow by the neutral one throughout training), the pigeons were not fooled and continued to recognize and avoid the hostile person. The quantity measured is difference in number of pigeons at the neutral site minus the hostile site. With \(n=32\) measurements, the mean difference in number of pigeons is 3.9 with a standard deviation of 6.8 . Test to see if this provides evidence that the mean difference is greater than zero, meaning the pigeons can recognize faces (and hold a grudge!)

In a survey of 1000 US adults, twenty percent say they never exercise. This is the highest level seen in five years. \(^{5}\) Find and interpret a \(99 \%\) confidence interval for the proportion of US adults who say they never exercise. What is the margin of error, with \(99 \%\) confidence?

Use the t-distribution to find a confidence interval for a mean \(\mu\) given the relevant sample results. Give the best point estimate for \(\mu,\) the margin of error, and the confidence interval. Assume the results come from a random sample from a population that is approximately normally distributed. A \(99 \%\) confidence interval for \(\mu\) using the sample results \(\bar{x}=88.3, s=32.1,\) and \(n=15\)
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