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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

Use the t-distribution to find a confidence interval for a difference in means \(\mu_{1}-\mu_{2}\) given the relevant sample results. Give the best estimate for \(\mu_{1}-\mu_{2},\) the margin of error, and the confidence interval. Assume the results come from random samples from populations that are approximately normally distributed. A \(90 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) using the sample results \(\bar{x}_{1}=10.1, s_{1}=2.3, n_{1}=50\) and \(\bar{x}_{2}=12.4, s_{2}=5.7, n_{2}=50 .\)

Survival in the ICU and Infection In the dataset ICUAdmissions, the variable Status indicates whether the ICU (Intensive Care Unit) patient lived (0) or died \((1),\) while the variable Infection indicates whether the patient had an infection ( 1 for yes, 0 for no) at the time of admission to the ICU. Use technology to find a \(95 \%\) confidence interval for the difference in the proportion who die between those with an infection and those without.

Assume the samples are random samples from distributions that are reasonably normally distributed, and that a t-statistic will be used for inference about the difference in sample means. State the degrees of freedom used. Find the proportion in a t-distribution above 2.1 if the samples have sizes \(n_{1}=12\) and \(n_{2}=12\).

Find a \(95 \%\) confidence interval for the mean two ways: using StatKey or other technology and percentiles from a bootstrap distribution, and using the t-distribution and the formula for standard error. Compare the results. Mean price of a used Mustang car online, in \$1000s, using data in MustangPrice with \(\bar{x}=15.98\), \(s=11.11,\) and \(n=25\)

As part of the same study described in Exercise 6.254 , the researchers also were interested in whether babies preferred singing or speech. Forty-eight of the original fifty infants were exposed to both singing and speech by the same woman. Interest was again measured by the amount of time the baby looked at the woman while she made noise. In this case the mean time while speaking was 66.97 with a standard deviation of \(43.42,\) and the mean for singing was 56.58 with a standard deviation of 31.57 seconds. The mean of the differences was 10.39 more seconds for the speaking treatment with a standard deviation of 55.37 seconds. Perform the appropriate test to determine if this is sufficient evidence to conclude that babies have a preference (either way) between speaking and singing.
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