Warm-up 9.1 Confidence Interval of the Mean. Answers to H.W. 8.2 E#26 – 32 and E#34 H A : The proportion of students wearing backpacks is not 60%.

Warm-up9.1 Confidence Interval of the Mean

Answers to H.W. 8.2 E#26 – 32 and E#34

HA: The proportion of students wearing backpacks is not 60%.

9.1 Confidence Interval for a Mean

S.E. for Proportion S.E. for Mean

95% C.I. for a Proportion 95% C.I. for the mean

97.8 98.0 98.2 98.2 98.2 98.6 98.8 98.8 99.2 99.4The mean body temperature, ,for this sample of ten

women is 98.52, and the standard deviation, s, is 0.527. What is the confidence interval for this sample?We don’t have the σ for the population. : (

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96.1

x

• Since the standard deviation varies greatly from sample to sample for the mean to calculate the interval one needs to use Student’s t.

• Student’s t was created by Gosset, who was a quality control engineer inGuinness Brewery in Dublin.

How to find when it is not known.

When the assumptions and conditions are met, we cancalculate the confidence interval.

. The critical value t*n -1 depends on the

confidence interval you specify and on the degrees of freedom (n-1), which you get from the sample size.

)(1* ySEty n

n

sySE )(

Or use invT( proportion, df)

Finding the confidence interval from a list of data97.8 98.0 98.2 98.2 98.2 98.6 98.8 98.8 99.2 99.4

Finding the confidence interval from the mean and s.d. of a sampleWhenever you construct a confidence interval based on t, there arethree conditions you must check. Officially, you need a random sample (or random assignment of treatments to units) an approximately normally distributed population or a large enough sample size that thedistribution of the sample mean is approximately normal in the case ofa survey, the size of the population should be at least ten times the sizeof the sample. (Listed in detail on pg 569 of textbook)

Quiz Directions • For confidence intervals and significance tests you

must show the formula, even if you use your calculator.

• No extra time given due to vocabulary• Read directions carefully.• Except for vocabulary answer everything in

complete sentences.When you finish work on the homework.

9.1 P#1-5

Warm-upDay 2 of 9.1

Copy the statements and fill in the missing word(s).1. is the formula for ___________________.

2. When completing a one-proportion z test, you calculate the p-value which informs you whether your p-hat is ____________________ .

3. When writing the hypotheses, the _____________________ is the first one to be considered. The ___________________ is the second one to be considered if the first one is rejected.

4. As sample size increases ____________________ decreases.

Word bank: level of confidence, margin of error significance test for a proportion, statistically significant, null hypothesis, alternate hypothesis, test statistic, p-value, critical values, level of significance

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b. Using the standard deviation for the sample is not alwaysaccurate since it changes based on sample size. Smaller samples will be less accurate and have a larger confidenceinterval. P2. a. 2.262 b. 2.447 c. 3.106 d. 2.920e. 2.695 f. 2.637On calculator a. invT(0.025,9), b. invT(0.025, 6) c. invT(0.005, 11) d. invT(0.05, 2) e. invT(0.005, 43)f. invT(0.005, 81)

P3. a. t* =3.182, t* ∙

The interval is 27 + 19.092, or 7.908 to 46.092

b. t* = 2.306, t* ∙

The interval is 6 + 2.306, or 3.694 to 8.306

c. t* = 2.131, t*

The interval is 9 + 25.572, or -16.572

n

s

n

s

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s

Or use InvT and not the table to get 1.984invT(.025, 121) = 1.984

9.1 (day 2) confidence interval of the mean

Gosset’s model uses the sample standard deviation toapproximate the confidence interval using t* (the student’s t)

A one-sample t-interval for the meanIn 2004, a team of researchers published a study ofcontaminants in farmed salmon. The study expressedconcerns over level of contaminants found. One contaminantfound is mirex. In parts per million an average of 0.0913 ppmwere found with a std. dev. of 0.0495 ppm in a sample of 150randomly selected salmon from a particular pond. The distributionof mirex concentration found in the salmon was approximatelynormal.

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n

sySE

ySE

yt n

Find the 95% confidence interval.

Interpret the meaning of the confidence interval inthe context of the problem.

ppm0495.0ppm0913.0150 syn

T-distribution curve

• Notice how the t-distribution curve changes based on the degrees of freedom.

Another exampleSome students are concerned about safety near anelementary school. Though there is a 15 MPH SCHOOL ZONEsign nearby, most drivers seem to go much faster than that,even when the warning sign flashes. The students randomlyselect 20 cars passing during the flashing zone times, andrecorded the averages. They found that the overall speedduring the flashing school zone times for the 20 time periodswas 24.6 mph with a std. dev. of 7.24 mph. Find the 90%confidence interval for the average speed of all vehiclespassing the school during those hours. A histogram of thedata demonstrates the distribution of the average speeds areapproximately normal.

9.2 One-sample t-test

A recent report on the evening news stated that teens watchan average of 13 hours of TV per week. A teacher at CentralHigh School believes that the students in her school actuallywatch more than 13 hours per week. She randomly selects 25students from the school and directs them to record their TVviewing hours for one week. The 25 students reported thefollowing number of hours.

Step 1: Check conditions and state test.

Step 2: State hypotheses.

n = 25 avg. =____ std. dev. = ___Step 3: To sketch the t-distribution curve accurately follow the following steps.

Homework: 9.1 E#2 – 6 and #10 and 12Bring textbook for textbook review