This week we do something useful with our z-values by using them to:

• CreateCONFIDENCE INTERVALSaround the mean, and
• TEST HYPOTHESESto see just how "sure" we are that they are statistically supported

CONFIDENCE INTERVALS

The z-values can be used to construct a confidence interval for a PROPORTION (%)If TWO conditions about the sampling distribution and sample size are met:

1. The Sampling Distribution must be a Normal (bell) Distribution (histogram bell-shaped?), or you can Multiply the sample size times the proportion probability (np), then n times the alternate probability n (1 - p) and IF BOTH are 5 or greater, it's a Normal Distribution.
2. The Sample Size (n) must be 30 or greater.

Now let's actually calculate a Confidence Interval around a Population mean, based on SAMPLE data.

1)The 896 residents of a retirement community were tested for Covid-19 and 120 tested positive. What is the 95% confidence interval around the TRUE proportion infected? Fill in the blanks:

n = _____, sample proportion infected (p-hat) = ________; sampling proportion not infected (q-hat) = ______

= _____ and /2 = ______ so, the Critical +z-value (z_c) = _________ ;

a) Calculate the Margin of Error boundary,E= z_c* SQRT [( p * q)/n] = ________ (don't forget this formula !)

b) Calculate the Confidence Interval: p-hat - E< population proportion (P)

c) State what this means:

2)(made-up problem)PROPORTION problem (use z)

35000 students took Stat 200 and 29000 passed. At the 1% significance level, what is the confidence interval around this proportion?

a) x = _______; n = ____________; p-hat =___________; q-hat = ___________; z_c= ____________

b) Using the formula for E, what is this margin of error?______________ (show setup and final value)

c) What is the Confidence Interval? _________ < P < ______________

NOW,what if either or both of the two conditions required to use the critical z-values are NOT met?

We use t- values. Unlike z-values,t-values depend on sample size (n)but if "n" gets up around 1000, the t-distribution looks more and more like the z-distribution (the Normal bell curve) and the z-values and t-values are almost equal:

"df" refers to "degrees of freedom" and is simply (n - 1).

See the Additional Guidance Attachment for more detailed instructions on using at-Table.

YOUR TURN:

3)(made-up data) Twenty-six(26) residents in a residential development were surveyed as to how many days the "bread-winner(s)" had been laid off due to Covid-19. The sample mean (x-bar) was 31 and the sample SD was 4. Construct an = 5% CI around the population mean based on this sample data.

a) Why is this a t- problem?_____________________________________

b) what is the critical t value? ________

c) Show the setup and calculate E: ______________________,

d) What is the = 5% CI: __________ < < _____________

So much for the easy part of this week's work - Let's move on to HYPOTHESIS TESTING

Follow the steps described in the Additional Guidance Attachment to this Homework

4) (Made up data) Persons over age sixty sway more than younger persons. This sway is GREATER than 22 mm (sway are the x-values).We want to test this hypothesis at the 5% level ( = 5%, so 95% confidence), 25 persons with ages varying between 61 and 78 were randomly sampled. The sample mean (x-bar) = 33 and the sample std dev (SD) = 13 We do NOT know the Population's standard deviation.

a) Is this a z-Test or a t-Test? Why?

b) Null hypothesis: Ho: ? _____ and the Alternate hypothesis Ha: ? _____

c) Based on your Ha, Is this a one-tailed or two-tailed test? ________

d) Calculate the TEST Statistic: t-Test = ( x-bar -) / [sample SD / sqrt(n) ] = (show setup then final result)

e) What is the CRITICAL t-value for = 0.05 with (n-1) degrees of freedom? (Use the t-Table)

f) Is the t-Test further (more standard deviations) into the right or left tail than the t-Critical? Which tail?

g) What is the p-value (from software)? Is this p-value larger or smaller than or alpha significance level of 2.5%?

h) Does the comparison of t-Test to t-Critical lead to the same conclusion as the p-value?

(hint:it MUST)

(Note: Since this is a greater-than situation we want the area to the right, so t-test would have to be positive and greater than t-critical to reject Ho)

i) What IS your conclusion about Ho? Do the elderly appear to sway more? (Keep in mind that this analysis does NOT prove they do, it just supports other sound-science findings, and it certainly does NOT tell anything about WHY the elderly might sway more.)

5) (made-up data) In 2008, there were 14,857 children in Arizona out of 32,601 who were diagnosed with Autism Spectrum Disorder (ASD) ("Autism and developmental," 2008).Nationally 1 in 88 children are diagnosed with ASD ("CDC features -," 2013). Is there sufficient data to show that the incidence of ASD is more in Arizona than nationally?

Test at the 1% level.

a) Is this a z-value or t-value problem?WHY?

b) What is the z or t-Test statistic formula?

c) Show your setup for this formula:

d) what is your calculated Test statistic value?

e) What is the Critical statistic value?

f) Is the Test greater or less than the Critical - hence do we accept or reject Ho?

g) What is the software-generated p-value? Is this p-value more or less than the alpha we chose? Do we accept or reject Ho based on this comparison? [must agree with answer to (e) ]

6)(made-up data) Economic dynamism, which is the index of productive growth in dollars for countries that are designated by the World Bank as middle-income is in the Table below. Countries that are considered high-income have a mean economic dynamism of53.3 Do the data show that the mean economic dynamism of middle-income countries is less than the mean for high-income countries? Test at the 5% level.

Table: Economic Dynamism of Middle-Income Countries (made up)

27.8	37.4	51.9	43.7	47.8	43.7
41.2	38.1	37.7	39.6	42.0	48.6
49.1	61.9	41.9	44.9	46.1	48.4
50.9	59.2	39.6	33.6	26.7	62.0

a) Is this a z-value or t-value problem?WHY?What are the SAMPLE Mean and Standard Deviation?

b) What is the z or t-Test statistic formula?

c) Show your setup for this formula:

d) what is your calculated Test statistic value?

e) What is the Critical statistic value?

f) Is the Test greater or less than the Critical - hence do we accept or reject Ho?

g) What is the software-generated p-value? Is this p-value more or less than the alpha we chose? Do we accept or reject Ho based on this comparison? [must agree with answer to (e) ]

To copy a Table from Word to Excel, first, highlight the entire Table in Word then right-click and "Copy" or hit Control C. Then, in Excel highlight the SAME number of columns across that the Table has (the above Table has 6 columns). Then right-click "Paste" or Control V to get the Table into Excel.If you just highlight one cell in Excel, it likely won't copy or will put all the values in this single column.

Once the Table values are in Excel, you can highlight the entire Table and then select the Function (f) you want to get the mean and standard deviation.