**Testing Population Mean with Unknown Standard Deviation**This page explains how to test a population mean when the standard deviation is unknown using the t-test. We will use the given data to determine the test statistic and the P-value, then make a decision regarding the null hypothesis.### Hypotheses and Significance LevelYou wish to test the following hypothesis at a significance level of α = 0.01:\[ H_0: \mu = 73.7 \]\[ H_a: \mu \neq 73.7 \]### Sample InformationThe population is assumed to be normally distributed, but the standard deviation is unknown. Your sample has the following characteristics:- **Sample size** ($n$): 54- **Sample mean** ($\bar{M}$): 65.8- **Sample standard deviation** ($SD$): 20.1### Test Statistic CalculationWhat is the test statistic for this sample? (Round the answer to 3 decimal places.)\[ \text{test statistic: } t = \_\_\_\_\_ \]### P-value CalculationWhat is the P-value for this sample? (Round the answer to 3 decimal places.)\[ \text{P-value: } \_\_\_\_\_ \]The P-value is:- ○ less than (or equal to) α- ○ greater than α### DecisionBased on the comparison between the P-value and α, decide:- ○ reject the null- ○ accept the null- ○ fail to reject the null### ConclusionSo, the final conclusion is:- ○ The data do **not** support the claim of the alternative hypothesis that the population mean is not equal to 73.7.- ○ The sample data **support** the claim of the alternative hypothesis that the population mean is not equal to 73.7.By completing the steps above, you will be able to determine if there is enough evidence to reject the null hypothesis in favor of the alternative hypothesis, based on your sample data.

Name: **Testing Population Mean with Unknown Standard Deviation**This page
Uploaded: 2024-03-20T07:06:16.000Z
Channel: Bartleby
Description: **Testing Population Mean with Unknown Standard Deviation**This page explains how to test a population mean when the standard deviation is unknown using the t-test. We will use the given data to determine the test statistic and the P-value, then mak

Introduction: Denote μ as the true population mean.

Math

Statistics

You wish to test the following claim (Ha) at a significance level of a = 0.01. H.:µ = 73.7 Ha:µ + 73.7 You believe the population is normally distributed, but you do not know the standard deviation. Your sample has: size: n = 54 mean: M = 65.8 standard deviation: SD = 20.1. What is the test statistic for this sample? (Round the answer accurate to 3 decimal places.) test statistic: t = What is the P-value for this sample? (Round the answer accurate to 3 decimal places.) P-value = The P-value is... less than (or equal to) a greater than a This leads to a decision to... O reject the null accept the null O fail to reject the null So, the final conclusion is that... O The data do not support the claim of the alternative hypothesis that the population mean is not equal to 73.7. O The sample data support the claim of the alternative hypothesis that the population mean is not equal to 73.7.

You wish to test the following claim (Ha) at a significance level of a = 0.01. H.:µ = 73.7 Ha:µ + 73.7 You believe the population is normally distributed, but you do not know the standard deviation. Your sample has: size: n = 54 mean: M = 65.8 standard deviation: SD = 20.1. What is the test statistic for this sample? (Round the answer accurate to 3 decimal places.) test statistic: t = What is the P-value for this sample? (Round the answer accurate to 3 decimal places.) P-value = The P-value is... less than (or equal to) a greater than a This leads to a decision to... O reject the null accept the null O fail to reject the null So, the final conclusion is that... O The data do not support the claim of the alternative hypothesis that the population mean is not equal to 73.7. O The sample data support the claim of the alternative hypothesis that the population mean is not equal to 73.7.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

See similar textbooks

Related questions

Q: You wish to test the following claim (H) at a significance level of a Ho: μ = 77.1 H :μ 7 77.1 You…

A: Given: The sample data is : 50.8,44,50 The hypothesis is given by; H0:μ=77.1Ha:μ≠77.1 The…

Q: A group of students take a Statistics Exam where the average was M = 85 and the standard deviation…

A: We have given that X~N( mu, sigma^2) mu=85 , sigma =6.8 ,X=85 Z-score =(x- mu)/sigma

Q: The quantitative data set under consideration has roughly a bell-shaped distribution. Apply the…

Q: You wish to test the following claim (Ha) at a significance level of a = 0.05. H,:µ = 55.6 H:u 55.6…

A: Given, Sample size = 14 Sample mean = 58.4 Sample standard deviation = 11.6 Population mean = 55.6…

Q: You wish to test the following claim (Ha) at a significance level of a = 0.02. H.:µ = 51.4 Ha:µ #…

A: Given,sample size(n)=20mean(M)=49.3standard deviation(S)=9.5H0:μ=51.4Ha:μ≠51.4α=0.02

Q: Pulse rates of women are normally distributed with a mean of 77.5 beats per minute and a standard…

A: Given data pulse rates of women are normally distributed with a mean of 77.5 beats per minute and…

Q: You wish to test the following claim (H.) at a significance level of a = 0.001. H.:µ = 54.8 Ha:µ >…

A: As the population standard deviation is not known, we will use a t-test. The t-statistic is computed…

Q: Based on what you have studied in this unit, reflect and answer the following questions: What are…

A: Step 1:The main characteristics of normal distribution are,It has same value of mean, median and…

Q: You wish to test the following claim (Ha) at a significance level of a = 0.002. H,:µ = 68.3 Ha:p +…

Q: You wish to test the following claim (H.) at a significance level of a 0.01. H.:µ = 50.5 Ha:u < 50.5…

A: From the provided information, Sample size (n) = 661 Sample mean (x̅) = 49.2 Sample standard…

Q: group of students take a Statistics Exam where the average was M = 85 and the standard deviation was…

A: We have given that X~N( mu , sigma^2) Mu= 85 ,sigma =6.8 Z-score = (x-mu )/sigma

Q: fou wish to test the following claim (Ha) at a significance level of a = 0.02. Ho:µ = 59.2 Haip >…

Q: You wish to test the following claim (Ha) at a significance level of a = 0.10. %3D Hại µ = 72.8 You…

A: Given : Sample mean (M)=75.9 Sample size(n)=12 Sample standard deviation (SD)=13.3 Significance…

Q: You wish to test the following claim (Ha) at a significance level of = 0.001. H.:µ = 67.4 Ha:µ >…

Q: You wish to test the following claim (Ha) at a significance level of a = 0.001. H.:µ = 65.4 Ha:4 +…

Q: bu wish to test the following claim (H.) at a significance level of a = 0.02. H.:µ = 53.1 Ha:µ +…

A: Given μ=53.1, n=52, M=47.2, SD=s=20.2

Q: What percentage of the population lands between the raw scores of 82 and 107?

A: Let , X→N(μ=M=85 , σ=6.8) Our aim is to find the percentage of the population lands between the raw…

Q: g claim (Ha) at a significance level of a = s normally distributed, but you do not kno of data: this…

A: Solution : For the given data : sample size, n = 5 sample mean, x̄ = 112.58 sample…

Q: You wish to test the following claim (Ha) at a significance level of a = 0.005. H,:µ = 67.9 Ha:u <…

A: Given n=124 M=67.1 S=5.9

Q: You wish to test the following claim (Ha) at a significance level of a = 0.10. H.:µ = 75.2 Ha:µ >…

A: The random variable X follows normal distribution. We have to test whether the population mean is…

Q: The blood platelet counts of a group of women have a bell-shaped distribution with a mean of 257.3…

A: The mean is 257.3 and the standard deviation is 64.1.

Q: You wish to test the following claim (H.) at a significance level of a = 0.02. H.:p = 61.6 Ha:> 61.6…

Q: IQ scores have a bell-shaped distribution with a mean of 100 and a standard deviation of 10. What…

Q: A group of students take a Statistics Exam where the average was M = 85 and the standard deviation…

A: The Normal distribution is also called as the Gaussian distribution. It is the most common…

Q: You wish to test the following claim (Ha) at a significance level of a = Ho: :μ = 90.8 Ha : μ + 90.8…

A: Hypothesis Test: Hypothesis testing can is a statistical tool that is used to identify if the…

Q: The weight of male babies less than 2 months old in the United States is normally distributed…

A: LetX= Weight of male baby less than two months oldSo given thatX∼N(μ=12,σ=3.1) To find the…

Q: 6. Bellow is an output from Excel One-Sample Z: Test of mu = 99 vs > 99 The assumed standard…

A: Given N=12, Mean=100.039, standard deviations σ=2.5

Q: You test the null hypothesis Ho : µ = 17 against the alternative hypothesis Ha: µ ≠ 17. You use the…

A: From the given information, It is provided to us that: H0:μ=17H1:μ≠17 It is two tailed test. Test…

Q: A popcorn company find that their 1kg bags have a mean of 985 grams with a standard deviation of 24…

Q: You wish to test the following claim (Ha) at a significance level of a = 0.005. H.:µ = 64.1 Ha:µ +…

A: Here we will use t- test statistic as sample size is less than 30 and we do not know the population…

Q: You wish to test the following claim (Ha) at a significance level of a = 0.005. H.:µ = 88.5 H.:µ >…

Q: You wish to test the following claim (Ha) at a significance level of a = 0.02. H.:µ = 90.2 a :H7…

A: Using the given data Find Test statistic = ? P value = ?

Q: H.:µ = 62.5 Ha:u > 62.5 %3D You believe the population is normally distributed, but you do not know…

Q: You wish to test the following claim (H.) at a significance level of a = 0.02. H.:µ = 78.8 Ha:u #…

Concept explainers

Inverse Normal Distribution

The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.

Mean, Median, Mode

It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.

Z-Scores

A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.

Topic Video

Question

**Testing Population Mean with Unknown Standard Deviation**

This page explains how to test a population mean when the standard deviation is unknown using the t-test. We will use the given data to determine the test statistic and the P-value, then make a decision regarding the null hypothesis.

### Hypotheses and Significance Level

You wish to test the following hypothesis at a significance level of α = 0.01:
\[ H_0: \mu = 73.7 \]
\[ H_a: \mu \neq 73.7 \]

### Sample Information

The population is assumed to be normally distributed, but the standard deviation is unknown. Your sample has the following characteristics:
- **Sample size** ($n$): 54
- **Sample mean** ($\bar{M}$): 65.8
- **Sample standard deviation** ($SD$): 20.1

### Test Statistic Calculation

What is the test statistic for this sample? (Round the answer to 3 decimal places.)

\[ \text{test statistic: } t = \_\_\_\_\_ \]

### P-value Calculation

What is the P-value for this sample? (Round the answer to 3 decimal places.)

\[ \text{P-value: } \_\_\_\_\_ \]
The P-value is:
- ○ less than (or equal to) α
- ○ greater than α

### Decision

Based on the comparison between the P-value and α, decide:

- ○ reject the null
- ○ accept the null
- ○ fail to reject the null

### Conclusion

So, the final conclusion is:

- ○ The data do **not** support the claim of the alternative hypothesis that the population mean is not equal to 73.7.
- ○ The sample data **support** the claim of the alternative hypothesis that the population mean is not equal to 73.7.

By completing the steps above, you will be able to determine if there is enough evidence to reject the null hypothesis in favor of the alternative hypothesis, based on your sample data.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Centre, Spread, and Shape of a Distribution$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, statistics and related others by exploring similar questions and additional content below.

Similar questions

You wish to test the following claim (H) at a significance level of a = 0.05. H.:µ = 65.2 Ha:µ < 65.2 You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n = 571 with mean M = 63.2 and a standard deviation of SD = 20.3. What is the test statistic for this sample? (Report answer accurate to three decimal places.) test statistic = What is the p-value for this sample? (Report answer accurate to four decimal places.) p-value = The p-value is. less than (or equal to) a O greater than a This test statistic leads to a decision to. reject the null accept the null fail to reject the null As such, the final conclusion is that. O There is sufficient evidence to warrant rejection of the claim that the population mean is less than 65.2. O There is not sufficient evidence to warrant rejection of the claim that the population mean is less than 65.2. O The sample data support the claim that the population mean is less than 65.2.…
Consider the following data: x 4 5 6 7 8 P(X = x) 0.2 0.2 0.1 0.2 0.3 Step 3 of 5: Find the standard deviation. Round your answer to one decimal place. Copy Data
Please help me to explain this questions in steps. Thank you.
i need help solving this homework question: the lengths of nails (in millimeters) at a certain plant are normally distributed with a mean of 20.00 and a standard deviation of 0.21. nails produced will be rejected unless their lengths are between 19.71 mm and 20.42 mm. what percent of the nails are accepted?
You wish to test the following claim (H) at a significance level of a = 0.01. Ho:μ = 87.4 Ha:μ< 87.4 You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n = 19 with mean M = 85.3 and a standard deviation of SD = 15.6. What is the p-value for this sample? (Report answer accurate to four decimal places.) p-value = The p-value is... O less than (or equal to) a Ogreater than a This p-value leads to a decision to... O reject the null accept the null O fail to reject the null
You wish to test the following claim (Ha) at a significance level of a = 0.005. H.: µ = 60.8 H.:u > 60.8 You believe the population is normally distributed, but you do not know the standard deviation. You obtain the following sample of data: data 92.4 61.2 97.2 82.3 What is the test statistic for this sample? (Report answer accurate to three decimal places.) test statistic = What is the p-value for this sample? (Report answer accurate to four decimal places.) p-value = The p-value is... less than (or equal to) a O greater than a This test statistic leads to a decision to... o reject the null o accept the null o fail to reject the null As such, the final conclusion is that... There is sufficient evidence to warrant rejection of the claim that the population mean is greater than 60.8. There is not sufficient evidence to warrant rejection of the claim that the population mean is greater than 60.8. The sample data support the claim that the population mean is greater than 60.8. There is…
In a particular statistics exam, the 35th percentile of final exam score is 42, and the 70th percentile of final exam score is 76. A histogram of the scores approximately follows the normal curve. What is the approximate standard deviation of the final exam scores? Hints: What is the z value in the normal table that gives you closest to 35% of the area between -z and z? Now what does that tell you about the number of SDs in between 42 inches and 76 inches? You might find it helpful to draw a picture of the normal curve with original units and standard units. Group of answer choices a. 7.5 b. 13.1 c. 20.7 d. 37.8
You wish to test the following claim (H.) at a significance level of a = 0.01. H.:µ = 75.8 Ha:µ > 75.8 You believe the population is normally distributed, but you do not know the standard deviation. Your sample has: size: n = 73 mean: M = 77.2 standard deviation: SD = 8.5. What is the test statistic for this sample? (Round the answer accurate to 3 decimal places.) test statistic: t = What is the P.value for this sample? (Round the answer accurate to 3 decimal places.) P-value = The P-value is... O less than (or equal to) a O greater than a This leads to a decision to... O reject the null O accept the null O fail to reject the null So, the final conclusion is that... O The data do not support the claim of the alternative hypothesis that the population mean is greater than 75.8. O The sample data support the claim of the alternative hypothesis that the population mean is greater than 75.8.
You wish to test the following claim (H.) at a significance level of a = 0.02. H.:µ = 71.8 Ha:u > 71.8 You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n = 19 with mean I = 89.2 and a standard deviation of s = 17.6. What is the p-value for this sample? (Report answer accurate to four decimal places.) p-value = The p-value is... O less than (or equal to) a O greater than a This p-value leads to a decision to... O reject the null O accept the null O do not reject the null As such, the final conclusion is that... O There is sufficient evidence to warrant rejection of the claim that the population mean is greater than 71.8. O There is not sufficient evidence to warrant rejection of the claim that the population mean is greater than 71.8. O The sample data support the claim that the population mean is greater than 71.8. O There is not sufficient sample evidence to súpport the claim that the population mean is greater…
You wish to test the following claim (H.) at a significance level of a 0.02. H,:p = 79.3 H: u > 79.3 You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n = 7 with mean M 105.5 and a standard deviation of SD= l6.5. %3D What is the p-value for this sample? (Report answer accurate to four decimal places.) p-value = The p-value is... O less than (or equal to) a greater than a This p-value leads to a decision to... reject the null Oaccept the null fail to reject the null As such, the final conclusion is that... 5There is sufficient evidence to warrant rejection of the claim that the population mean is greater than 79.3. | There is not sufficient evidence to warrant rejection of the claim that the population mean is greater than 79.3. The sample data support the claim that the population mean is greater than 79.3. There is not sufficient sample evidence to support the claim that the population mean is greater than 79.3.
You wish to test the following claim (H) at a significance level of a = 0.10. H.:µ = 66.2 Ha:µ + 66.2 You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n = 10 with mean M = 43.4 and a standard deviation of SD = 18.5. What is the test statistic for this sample? (Report answer accurate to three decimal places.) test statistic = What is the p-value for this sample? (Report answer accurate to four decimal places.) p-value = The p-value is. O less than (or equal to) a greater than a This test statistic leads to a decision to... O reject the null O accept the null O fail to reject the null As such, the final conclusion is that. O There is sufficient evidence to warrant rejection of the claim that the population mean is not equal to 66.2. O There is not sufficient evidence to warrant rejection of the claim that the population mean is not equal to 66.2. O The sample data support the claim that the population mean is not…
You wish to test the following claim (H,) at a significance level of a = 0.01. H.:p=53.3 Ha:p > 53.3 You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n 5 with mean M = 59.8 and a standard deviation of SD= 6.9. eranra What is the p-value for this sample? (Report answer accurate to four decimal places.) p-value =