Calculate the sample mean directly from the frequency table you got in Part (c)

MTH 1210

Exam I

February 18, 2021

Name:

 

 

Notation

n = sample size, x̄ = sample mean, s = sample stdev. N = Population size, µ = population mean, σ = population stdev. Qi the ith quartile,

Descriptive Measures

Chapter 3 Sample mean: x̄ = Σxi

n

Sample Standard Deviation: s = √

Σx2 i −(Σxi)2/n n−1 or s =

√ Σ(xi−x̄)2

n−1

Range = Max – Min

Interquartile Range: IQR = Q3 − Q1 Lower Limit: L.L. = Q1 − 1.5 · IQR, Upper Limit: U.L. = Q3 + 1.5 · IQR Population Mean: µ = Σxi

N

Population Standard Deviation: σ = √

Σx2 i

N − µ2 or σ =

√ Σ(xi−µ)2

N

Standardized variable (Z-score): z = x−µ σ

Descriptive Methods in Regression and Correlation

Chapter 14 Sxx,Sxy, and Syy:

Sxx = Σ(xi − x̄)2 or Sxx = Σx2i − Σ(xi) 2/n

Sxy = Σ(xi − x̄)(yi − ȳ) or Sxy = Σ(xi)(yi) − Σ(xi)Σ(yi)/n

Syy = Σ(yi − ȳ)2 or Syy = Σy2i − Σ(yi) 2/n

Regression Equation: ŷ = b0 + b1x, where

b1 = Sxy Sxx

and

b0 = 1 n (Σyi − b1Σxi) or b0 = ȳ − b1x̄

Total Sum of Squares: SST = Syy Regression Sum of Squares: SSR = (Sxy)2/Sxx

Error Sum of Squares: Syy − (Sxy)2/Sxx Coefficient of Determination: r2 = SSR

SST

Linear Correlation Coefficient: r = Sxy√ SxxSyy

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1. 46 tribble ranchers were questioned on current herd size. The sum of all the heard sizes is Σxi = 2760. Part of the sorted data is shown here:

5, 7, 11, 13, 15, 17, 18, 18, 19, 21, 24, 27, 27, 27, 31, · · ·

· · · , 58, 61, 62, 63, 65, 65, 65, 66, 67, 69, 71, 71, 71, 92, 93, 204, 255

(a) Find the IQR of this data set. Refer to the entire data set, including the values not shown. (do not make up values for the missing middle section of the data)

(b) Find the 5% trimmed mean. (Trimming 5% from each end of the entire data set).

2. Data for number of Air Kracken spotted above Denver for the years 1980 to 2020 yielded the following 5-number summary: {19, 39, 46, 51, 65}.

(a) Find the Upper Limit (UL), and determine if there are any potential outliers from above.

(b) Find the Lower Limit (LL), and determine if there are any potential outliers from below.

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3. Suppose you buy a new jetpack whose advertised mileage is 35 miles per gallon (mpg). After flying your jetpack for several months, you find that its mileage is 30.36 mpg. You telephone the manufacturer and learn that the standard deviation of gas mileages for all cars of the model you bought is 1.15 mpg.

(a) Find the z-score for the gas mileage of your jetpack, assuming the advertised claim is correct.

(b) Does it appear that your jetpack is getting unusually low gas mileage? Explain your answer.

4. A recent study found the mean x̄ number of social media memberships of college students is 5 with a standard deviation of σ = .95. Becky belongs to 6 social media sites, calculate her z-score.

5. A curious student went around the math department and asked the professors how many tattoos they have when taken as a fraction of the total body surface area, (for instance a full back tattoo would be 1/4 body coverage) the student found that the median for the tattoo coverage was 3/10 but the mean was up at 3/5. What could explain this difference?

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6. A Tae Kwon Do dojong (school) has implemented a new training program to increase the speed that a student can kick. Students in the new program are tested by counting the number of double kicks they can throw in 30 seconds. The results are as follows:

106, 55, 94, 75, 89, 79, 68, 67, 78, 83, 78, 88, 87, 83, 69

(a) Find the mean, median, and mode

(b) draw a histogram for this data, be sure to label your bins.

(c) Find the 5 number summary.

(d) Draw a box plot

7. Give an example of a

(a) qualitative variable.

(b) discrete, quantitative variable.

(c) continuous, quantitative variable.

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8. ghotI’mey naS bIH norghmey’e’. translation Norgs are vicious fish.

Given the data: Norg length (x) ={110, 160, 89, 130, 98 } in cm Norg weight (y) = {46, 57, 32, 49 , 40} in Kg

(a) Find the regression equation that models norg weight as a function of length.

(b) What does the model predict for the weight of a 170 cm norg?

9. A researcher studied the weight of two species of targs: brown targs, and gray targs. Samples from both species yielded the following information: (the weight is in Killograms).

Species Mean Standard Deviation Brown targs 27.2 2.5 Gray targs 39.8 7.3

(a) What weight of a gray targ would make it comparable to a brown targ that weighs 26 Kg? (hint: compare z-scores)

(b) If a brown targ weighs 34.9 Kg, would you classify it as an outlying observation? (Justify your answer).

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10. The following graph Shows a very strong correlation between the number of pirates and global temperature.

The coefficient of determination has been calculated as r2 = .973.

Is this sufficient evidence to conclude the pirate population is the leading cause of global warming? (explain why or why not)

11. Given x̄ = 5, ȳ = 4, Σ(xi − x̄)2 = 466, Σ(xi − x̄)(yi − ȳ) = 346, and Σ(yi − ȳ)2 = 282 Find The regression equation, SST, SSR, SSE, and r

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Extra Credit Problems

1. The price of a jacket was checked in 16 clothing stores. The following data (in $) were recorded: 47, 103, 64, 25, 125, 33, 61, 42, 56, 136, 75, 41, 38, 47, 52, 71.

(a) Find the mean price.

(b) Find the median price.

(c) Present the data in a grouped frequency table, using the intervals 20-40, 40-60, …etc. Write the intervals and their frequencies, in the table below. (No need to show work here.) Intervals Frequency

(d) Calculate the sample mean directly from the frequency table you got in Part (c), as if the original data didn’t exist.(you can add extra columns next to the table above). (grouped frequency formula x̄ = Σ(xjfj)/n where xj is the midpoint of each bin, and fj is the frequency.)

(e) Why is the mean obtained from the frequency table a little different from the mean that you calculated from the raw data in Part (a)?

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