Statistics- Lesson 23 - One-Sample Interval Estimate of a Population Mean

1

Statistical Inference

Student’s Name

Institution Affiliation

Course Number: Course Name

Instructor’s Name

Assignment’s Due Date

2

Question 1

Confidence interval X = µ +/- zσ .Where Z is the Z score and σ is the standard error of the

mean.

The Standard Error =

At 99% the Z score is 2.576 according to the standard normal tables

The Confidence interval formula for X = µ +/- zσ .

Using Stats crunch software,

Sample size (n)=25, Sample mean=3.4548 and standard deviation=2.9682

Assuming the data is normally distributed,

Confidence interval X = µ +/- z *

Confidence interval X = 3.4548 2.576*( )

Confidence interval X = 3.4548 1.529= [1.9256, 4.9840]CO2 emissions

Round off to 3d.p the 99%confidence interval is between 1.926 and 4.984

The procedure for constructing a confidence interval estimate of a population mean when the Central Limit Theorem applies can be found on page 2 of the Lesson 23 notes. Remember that your choice of distribution depends partially on whether or not you know (or can guess) the value of σ, the population standard deviation. In this course, you should only use the normal distribution if specifically told what the value of σ is. If the symbol σ or the words "population standard deviation" do not appear in the question, then you should use the t distribution.

σ √n

σ √n

± 2.9682

√25

±

3

In this question, you are using the "With Summary" option. Make sure you enter the correct mean, standard deviation, and sample size into the proper fields, and that the confidence level is correct. Also, double check that your answers are rounded according to the instructions.

Question 2

Bootstrap used in small sample size and no assumptions of normal distribution

Confidence interval X = µ +/- t *

Using the software formula, the original data has a sample size of 20, sample mean of 3.02

and standard deviation 0f 1.211

Using a 10,000 resample and seed for Mersenne Twister of 12641, it results to a t=13.673,

p<.05

The mean difference is 3.702 and bias of -.00318 and standard error of .2626

The 95% bootstrap for one sample t-test is between 3.206 and 4.2295

Round off to 3d.p the 95% bootstrap confidence interval is between 3.206 and 4.230

The procedure for constructing a confidence interval estimate of a population mean using the bootstrap method can be found on pages 5-6 of the Lesson 23 notes. Make sure you select the correct data column to sample from, that you are using the fixed see 12641, and that you click the "1000 times" menu item exactly 10 times. For a 95% confidence interval, select the 2.5th percentile as your lower bound and the 97.5th percentile as your upper bound. Also, double check that your answers are rounded according to the instructions. Results for item 3. 3

σ √n

4

Question 3

Confidence interval X = µ +/- zσ .Where Z is the Z score and σ is the standard error of the

mean.

The Standard Error =

At 90% the Z score is 1.645 according to the standard normal tables

Confidence interval formula for X = µ z *

Males

Sample size (n)=40, Sample mean=66.825 and standard deviation=12.555

Assuming the data is normally distributed,

Confidence interval X = 66.825 1.645*( )

Confidence interval X = 66.825 3.266= [63.559, 70.091]beats per minute

Round off to 1d.p the 90%confidence interval is between 63.6 and 70.1 beats per minute

Females

Sample size (n)=40, Sample mean=76.225 and standard deviation=13.478

Assuming the data is normally distributed,

Confidence interval X = 76.225 1.645*( )

σ √n

± σ

√n

± 12.55

√40

±

± 13.478

√40

5

Confidence interval X = 76.225 3.506= [72.719, 79.731] beats per minute

Round off to 1d.p the 90%confidence interval is between 72.7 and 79.7 beats per minute

Question 4

d. The confidence intervals do not overlap, so it appears adult females have a

significantly higher mean pulse rate than adult males

The procedure for constructing a confidence interval estimate of a population mean when the Central Limit Theorem applies can be found on page 2 of the Lesson 23 notes. Remember that your choice of distribution depends partially on whether or not you know (or can guess) the value of σ, the population standard deviation. In this course, you should only use the normal distribution if specifically told what the value of σ is. If the symbol σ or the words "population standard deviation" do not appear in the question, then you should use the t distribution. In this question, you are using the "With Data" option. Make sure you select the correct data column when calcuating the confidence limits for each gender, and also that the confidence level is correct. Also, double check that your answers are rounded according to the instructions.

±