A random sample of 46 adult coyotes in a region of northern Minnesota showed the average age to be 2.07 years, with sample standard deviation years owve, RRI that the overall population mean age of coyotes is 1.75. Do the sample data indicate that oyotes in this region of northern Minnesota tend to live longer than the average of 1.75 years? Use a -0.01. A USE SALT (a) what is the level of significance? State the null and alternate hypotheses. O Họ 1.75 yr H L75 yr Hoi 1.75 yr: H<1.75 yr O Ho 1.75 yr; H>1.75 yr O Hại < 1.75 yr; H 175 yr O Ho: H> 1.75 yr: H:- 1.75 yr (b) what sampling distribution will you use? Explain the rationale for your choice of sampling distribution. O The Student's t, since the sample size is large and a is known. The standard normal, since the sample size is large and a is unknown. The standard normal, since the sample size is large and e is known The Student's t, since the sample size is large and a is unknown. What is the value of the sample test statistic? (Round your answer to three decimal places.) (e) Estimate the Pvalue. OPvalue 0.250 0.100 Pvalue <0.250 0.050
A random sample of 46 adult coyotes in a region of northern Minnesota showed the average age to be 2.07 years, with sample standard deviation years owve, RRI that the overall population mean age of coyotes is 1.75. Do the sample data indicate that oyotes in this region of northern Minnesota tend to live longer than the average of 1.75 years? Use a -0.01. A USE SALT (a) what is the level of significance? State the null and alternate hypotheses. O Họ 1.75 yr H L75 yr Hoi 1.75 yr: H<1.75 yr O Ho 1.75 yr; H>1.75 yr O Hại < 1.75 yr; H 175 yr O Ho: H> 1.75 yr: H:- 1.75 yr (b) what sampling distribution will you use? Explain the rationale for your choice of sampling distribution. O The Student's t, since the sample size is large and a is known. The standard normal, since the sample size is large and a is unknown. The standard normal, since the sample size is large and e is known The Student's t, since the sample size is large and a is unknown. What is the value of the sample test statistic? (Round your answer to three decimal places.) (e) Estimate the Pvalue. OPvalue 0.250 0.100 Pvalue <0.250 0.050
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Hypothesis Testing for Coyote Lifespan in Northern Minnesota
---
#### Problem Statement:
A random sample of 46 adult coyotes in a region of northern Minnesota showed the average age to be \( \bar{x} = 2.07 \) years, with a sample standard deviation \( s = 0.88 \) years. However, it is thought that the overall population mean age of coyotes \( \mu \) is 1.75 years. Do the sample data indicate that coyotes in this region of northern Minnesota tend to live longer than the average of 1.75 years? Use \( \alpha = 0.01 \).
#### Steps:
##### (a) Significance Level:
**What is the level of significance?**
- Given \(\alpha = 0.01\).
**State the null and alternate hypotheses:**
- \( H_{0}: \mu = 1.75 \text{ yr} \)
- \( H_{a}: \mu > 1.75 \text{ yr} \)
##### (b) Sampling Distribution:
**Which sampling distribution will you use? Explain the rationale for your choice of sampling distribution.**
1. **The binomial distribution**: Incorrect - This is for discrete data.
2. **The standard normal** since the sample size is large and \(\sigma\) is known: Not applicable here.
3. **The standard normal** since the sample size is large and \(\sigma\) is unknown: Incorrect - \(\sigma\) is unknown.
4. **The Student's \( t \)** since the sample size is large and \(\sigma\) is unknown: Correct choice - Even though the sample size is large, the population standard deviation \(\sigma\) is unknown.
**What is the value of the sample test statistic? (Round your answer to three decimal places.)**
\[
t = \frac{\bar{x} - \mu}{s / \sqrt{n}} = \frac{2.07 - 1.75}{0.88 / \sqrt{46}} = 2.567
\]
##### (c) P-Value Estimation:
**Estimate the P-value:**
- \( P \text{-value} \approx 0.250 \) (Just slightly more than 0.01 is too high for significance, suggesting potential considering secondary sources for more precision.)
- To better make the P](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1da81aa5-c0e9-4d6e-87eb-a5febb367169%2F72eb56c3-a666-496c-9d51-fe42cd4a7946%2Fefoqp8c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Hypothesis Testing for Coyote Lifespan in Northern Minnesota
---
#### Problem Statement:
A random sample of 46 adult coyotes in a region of northern Minnesota showed the average age to be \( \bar{x} = 2.07 \) years, with a sample standard deviation \( s = 0.88 \) years. However, it is thought that the overall population mean age of coyotes \( \mu \) is 1.75 years. Do the sample data indicate that coyotes in this region of northern Minnesota tend to live longer than the average of 1.75 years? Use \( \alpha = 0.01 \).
#### Steps:
##### (a) Significance Level:
**What is the level of significance?**
- Given \(\alpha = 0.01\).
**State the null and alternate hypotheses:**
- \( H_{0}: \mu = 1.75 \text{ yr} \)
- \( H_{a}: \mu > 1.75 \text{ yr} \)
##### (b) Sampling Distribution:
**Which sampling distribution will you use? Explain the rationale for your choice of sampling distribution.**
1. **The binomial distribution**: Incorrect - This is for discrete data.
2. **The standard normal** since the sample size is large and \(\sigma\) is known: Not applicable here.
3. **The standard normal** since the sample size is large and \(\sigma\) is unknown: Incorrect - \(\sigma\) is unknown.
4. **The Student's \( t \)** since the sample size is large and \(\sigma\) is unknown: Correct choice - Even though the sample size is large, the population standard deviation \(\sigma\) is unknown.
**What is the value of the sample test statistic? (Round your answer to three decimal places.)**
\[
t = \frac{\bar{x} - \mu}{s / \sqrt{n}} = \frac{2.07 - 1.75}{0.88 / \sqrt{46}} = 2.567
\]
##### (c) P-Value Estimation:
**Estimate the P-value:**
- \( P \text{-value} \approx 0.250 \) (Just slightly more than 0.01 is too high for significance, suggesting potential considering secondary sources for more precision.)
- To better make the P
Transcribed Image Text:**Understanding Sampling Distributions and P-Values**
To enhance your comprehension of hypothesis testing, we will explore the concept of sampling distributions and P-values using graphical representations. These explanations are integral for interpreting statistical significance.
### Graphical Representations
The provided image features four graphs, each illustrating a normal distribution (bell curve) with annotations on the horizontal axis ranging from approximately -4 to 4. Let's dissect each graph:
1. **Graph (Top Left)**:
- A standard normal distribution is displayed with its peak at zero.
- No specific area under the curve is shaded.
2. **Graph (Top Right)**:
- A standard normal distribution similar to the first, but with a shaded region extending symmetrically from approximately -2 to 2 on the horizontal axis.
- This shaded area represents the range of P-values within a certain significance level, indicating the probability of obtaining a test statistic at least as extreme as the one observed.
3. **Graph (Bottom Left)**:
- Again, a standard normal distribution centered at zero.
- Similar to the first graph, no specific area is shaded.
4. **Graph (Bottom Right)**:
- Displays a normal distribution centered at zero.
- A shading is present beyond approximately -2 and 2, highlighting the tails of the distribution.
- These tails represent the extreme values that would lead to rejection of the null hypothesis based on the P-value.
### Interpretation of Hypothesis Tests
The text under the graphs poses a question regarding hypothesis testing at various significance levels (α). The significance levels considered are 0.01 and 0.1.
#### Question:
**Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?**
#### Options:
1. **At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.**
- This implies that the observed data falls within regions (tails) that correspond to a P-value less than 0.01, leading to rejection of the null hypothesis.
2. **At the α = 0.1 level, we reject the null hypothesis and conclude the data are statistically significant.**
- Here, the data falls within regions that correspond to a P-value less than 0.1, indicating the data is statistically significant at this level.
3. **At the α =
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