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a) Males of the Costa Rican three-wattled bellbird are the loudest birds in the world. Their bell-like mating song can be heard from over half a mile away by humans. Juanita, your Costa Rican friend, saw a report online claiming that these birds can be as loud as 100 dB (decibels). To put it in perspective, a live rock concert makes around 100 dB of sound. While these birds can be quite loud, a hundred decibels seems like an exaggeration. Skeptical, Juanita decides to conduct a hypothesis test to decide whether the internet report can be trusted based on a random sample of 9 adult male three-wattled bellbirds. - Assume that the loudness of bellbird songs are approximately normally distributed with mean p (in dB) and variance o2 (in dB2) -both unknown ER (i) O Ho:p=100, HA:µS100. O Ho:p0100, HA:µ<100. Ο Haμ=100, HA:μ 1 0. O Ho:p=100, Ha:p<100. O Ho:us100, HA:p>100. (ii) The average loudness and variance for Juanita's sampled birds were computed to be 94.619 dB and 49.0001 dB2 respectively. Conduct an appropriate test to choose between the hypotheses you identified in (i), and report the p-value, and your decision (accept or reject Ho) at a 2% level of significance (that is, a = 0.02). (iii) Construct a 95% confidence interval for µ, and interpret it.

a) Males of the Costa Rican three-wattled bellbird are the loudest birds in the world. Their bell-like mating song can be heard from over half a mile away by humans. Juanita, your Costa Rican friend, saw a report online claiming that these birds can be as loud as 100 dB (decibels). To put it in perspective, a live rock concert makes around 100 dB of sound. While these birds can be quite loud, a hundred decibels seems like an exaggeration. Skeptical, Juanita decides to conduct a hypothesis test to decide whether the internet report can be trusted based on a random sample of 9 adult male three-wattled bellbirds. - Assume that the loudness of bellbird songs are approximately normally distributed with mean p (in dB) and variance o2 (in dB2) -both unknown ER (i) O Ho:p=100, HA:µS100. O Ho:p0100, HA:µ<100. Ο Haμ=100, HA:μ 1 0. O Ho:p=100, Ha:p<100. O Ho:us100, HA:p>100. (ii) The average loudness and variance for Juanita's sampled birds were computed to be 94.619 dB and 49.0001 dB2 respectively. Conduct an appropriate test to choose between the hypotheses you identified in (i), and report the p-value, and your decision (accept or reject Ho) at a 2% level of significance (that is, a = 0.02). (iii) Construct a 95% confidence interval for µ, and interpret it.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
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### Analyzing the Loudness of the Costa Rican Three-Wattled Bellbird #### Background Males of the Costa Rican three-wattled bellbird are considered to be the loudest birds globally. Their mating calls can be heard from over half a mile away by humans. A claim suggests that these birds can reach a loudness of 100 dB (decibels). To put this in perspective, a live rock concert typically reaches about 100 dB. To validate this claim, Juanita conducts a hypothesis test with a sample of 9 adult male three-wattled bellbirds. #### Hypotheses (i) Determine the null and alternative hypotheses: - \( H_0 \): μ = 100, \( H_A \): μ ≤ 100. - \( H_0 \): μ = 100, \( H_A \): μ < 100. - \( H_0 \): μ = 100, \( H_A \): μ ≥ 100. - \( H_0 \): μ = 100, \( H_A \): μ > 100. - \( H_0 \): μ ≤ 100, \( H_A \): μ > 100. #### Statistical Analysis (ii) **Hypothesis Testing** - The sample average loudness is 94.619 dB with a variance of 49.0001 dB². Conduct a test to choose between the hypotheses identified in part (i). Report the p-value and decision at a 2% significance level (α = 0.02). (iii) **Confidence Interval** - Construct a 95% confidence interval for μ and interpret the results. (iv) **Sample Size Determination** - Calculate the required sample size to ensure a 95% confidence interval with a margin of error at most 5 dB. (v) **Additional Scenario** - Assuming Juanita mistakenly treats her sample variance as the population variance, find her p-value for this test, and determine whether she accepts or rejects \( H_0 \) at the same significance level as in part (ii). This analysis helps assess the credibility of the loudness claim and teaches statistical techniques in hypothesis testing and confidence interval estimation.
Transcribed Image Text:### Analyzing the Loudness of the Costa Rican Three-Wattled Bellbird #### Background Males of the Costa Rican three-wattled bellbird are considered to be the loudest birds globally. Their mating calls can be heard from over half a mile away by humans. A claim suggests that these birds can reach a loudness of 100 dB (decibels). To put this in perspective, a live rock concert typically reaches about 100 dB. To validate this claim, Juanita conducts a hypothesis test with a sample of 9 adult male three-wattled bellbirds. #### Hypotheses (i) Determine the null and alternative hypotheses: - \( H_0 \): μ = 100, \( H_A \): μ ≤ 100. - \( H_0 \): μ = 100, \( H_A \): μ < 100. - \( H_0 \): μ = 100, \( H_A \): μ ≥ 100. - \( H_0 \): μ = 100, \( H_A \): μ > 100. - \( H_0 \): μ ≤ 100, \( H_A \): μ > 100. #### Statistical Analysis (ii) **Hypothesis Testing** - The sample average loudness is 94.619 dB with a variance of 49.0001 dB². Conduct a test to choose between the hypotheses identified in part (i). Report the p-value and decision at a 2% significance level (α = 0.02). (iii) **Confidence Interval** - Construct a 95% confidence interval for μ and interpret the results. (iv) **Sample Size Determination** - Calculate the required sample size to ensure a 95% confidence interval with a margin of error at most 5 dB. (v) **Additional Scenario** - Assuming Juanita mistakenly treats her sample variance as the population variance, find her p-value for this test, and determine whether she accepts or rejects \( H_0 \) at the same significance level as in part (ii). This analysis helps assess the credibility of the loudness claim and teaches statistical techniques in hypothesis testing and confidence interval estimation.
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