This means when we perform a 95% confidence interval 5% of all intervals will not contain the true parameter. Therefore, we assume a 5% risk we might get an interval that does not contain the true parameter. We hope we get one of the "good" intervals. In practice, we will not know. The simulation repeatedly samples from a population, calculates a confidence interval for each sample and indicates how many confidence intervals obtain the true mean.

The goal of this simulation is to visualize and validate the definition of a confidence interval.

Getting Started: Go to the Simulation in Lesson 25 in the Week 4 Module in Canvas.

1.Start with a 90% confidence interval and the population for standard deviation.

2.Change Sample Size to 15 and "# of Simulations" to 1.

3.This means you are just taking 1 sample of n = 15. This is most similar to what we do in "the real world". We only take one sample to estimate a parameter.

a.(1 point) Does your 90% confidence interval contain the true mean?

Yes

b.(1 point) Increase "# of Simulations" to 1000. Theoretically, 90% of the sample means we obtain should result in an interval that contains the true parameter. Does this seem to be the case?

Yes

c.(1 point) What type of sample will fail to capture the true parameter?

Decrease "# of Simulations" to 100. The intervals that don't contain the true mean are indicated in red. You can hover over a sample mean (dot in center of interval) to see it's value and the interval's margin of error.

Is there a common feature from the intervals that do not contain the true mean?

yes

Where are their sample means with respect to the sample means of the intervals that do contain the parameter?

Consider the placement of the sample mean in the sampling distribution.

Optional: Perform the previous steps using confidence levels 95% and 99%.

d.(1 point) How does sample size affect your confidence intervals?

Continue with a 90% Confidence Level and "# of Simulations" at 100.

Choose a smaller sample size between 2 and 10 observe the width of your intervals.

Increase the sample size to something between 30 and 100 observe the width of your intervals.

Increase your sample size to 1000 observe the width of your intervals.

It is the sample size that influences the confidence interval. The true size of population does not affect it. Confidence intervals from large samples tend to be quite narrow in width which results in more precise estimates. Increasing the sample decreases the width of confidence interval because it decreases the standard error.

e.(1 point) How does the confidence level affect your confidence intervals?

Continue with a 90% Confidence Level, "# of Simulations" at 100 and a moderate sample size between 30 and 100. Observe the width of your intervals.

Increase the confidence level to 95% observe your intervals.

Increase the confidence level to 99% observe your intervals.

Part 3: (3 points) Solving a basic hypothesis test for a mean.

Given the following information:

(1 point) Calculate the z test statistic and p-value.

Z= -1.72P=0.0854

(1 point) Draw the area under the the curve that corresponds to the p-value

oHint: There are many ways you can do this. In word, use "review" and "Start Inking".

oYou can draw it by hand on paper, take a picture and paste it below.

(1 point) Make a conclusion.

oState whether we should reject or fail to reject the null hypothesis based on the significance level

oMake a statement in terms of the alternative hypothesis.

Part 4. (9 points) A local bakery bakes thousands of loaves a bread per week. The bakery has a machine that automatically measures the amount of dough needed per loaf such that each baked loaf weighs 480 grams. Lately, the bakers have noticed that the number of loaves produced from this process has decreased. They are suspicious that the machine is measuring out too much dough per loaf and is therefore making fewer overall loaves. The bakery randomly samples 60 loaves of baked bread and measures the weight in grams. The sample yields a mean ofgrams. Assume .Use a significance level of 0.05

State: Is there evidence that the average weight of loaves is greater than 480 grams?

Plan:

a.(1 point) State the null and alternative hypotheses to answer the question of interest.

b.(1.5 points) Check conditions for inference. List the conditions and state whether they are met.

Solve:

c.(1 point) Calculate the test statistic. Show work.

d.(1 point) Draw the distribution for the test statistic and shade the region corresponding to the p-value. What is the p-value for the test?

e.(1 point) Calculate a 95% confidence interval for . Show work.

Conclude:

f.Write a four-part conclusion describing the results.

• (1 point) Provide a statement in terms of the alternative hypothesis.
• (0.5 point) State whether (or not) to reject the null.
• (1 point) Give an interpretation of the point and interval estimate.
• (1 points) Include context.