Question

In: Statistics and Probability

Data on the rate at which a volatile liquid will spread across a surface are in...

Data on the rate at which a volatile liquid will spread across a surface are in the table. Complete parts athrough

c.

Time(Minutes): 0,2,4,6,8,10,12,14,16,18,20,25,30,45,60

Mass (Pounds): 6.62, 5.96, 5.47, 4.85, 4.38, 4.05, 3.61, 3.09, 2.74 ,2.48, 2.23, 1.56, 0.94, 0.18, 0.00

Find a 98​% confidence interval for the mean mass of all spills with an elapsed time of 56minutes. Interpret the result.

What is the confidence​ interval?

(    ),(   )

​(Round to three decimal places as​ needed.)

Interpret the result. Choose the correct answer below.

A. We are 98​% confident that the interval will contain 56minutes.

B. We are 98​% confident that the interval will contain the mean mass of the spill before 56minutes has passed.

C. We are 98​% confident that the interval will not contain the mean mass of the spill at

56minutes.

D. We are 98​% confident that the interval will contain the mean mass of the spill after 56minutes.

b.Find a 98​% prediction interval for the mass of a single spill with an elapsed time of

56minutes. Interpret the result.

What is the prediction​ interval?

(.   ),(.  )

​(Round to three decimal places as​ needed.)

Interpret the result. Choose the correct answer below.

A.We are 98​% confident that the interval will contain the mass of the spill after 56minutes.

B.We are 98​% confident that the interval will contain 56minutes.

C.We are 98​% confident that the interval will contain the mass of the spill before 56minutes has passed.

D.We are 98​% confident that the interval will not contain the mass of the spill after 56minutes.

c.Compare the​ intervals, parts aand

b.

Which interval is​ wider? Will this always be the​ case? Explain. Fill in the blanks below.

The (prediction/confidence/neither) interval is wider. This (will/will not) always be the case because the error of this interval is the (random error/ sum of two errors/neither)

Solutions

Expert Solution

  • a)

    98% confidence interval: (-2.5793,0.3990)

    D We are 98​% confident that the interval will contain the mean mass of the spill after 56 minutes

  • b)

    98% prediction interval:(-2.2872,3.0851)

    A.We are 98​% confident that the interval will contain the mass of the spill after 56 minutes

    The prediction interval is wider. This will always be the case because the error of this interval Which interval is​ wider

  • option:neither

Calculation below using excel

No x=Time(Minutes) y=Mass (Pounds) (x-xbar)^2 (y-ybar)^2 (x-xbar)*(y-ybar)
1 0 6.62 324 11.623554 -61.368
2 2 5.96 256 7.5588338 -43.98933333
3 4 5.47 196 5.1045871 -31.63066667
4 6 4.85 144 2.6874138 -19.672
5 8 4.38 100 1.3673404 -11.69333333
6 10 4.05 64 0.7044804 -6.714666667
7 12 3.61 36 0.1594671 -2.396
8 14 3.09 16 0.0145604 0.482666667
9 16 2.74 4 0.2215271 0.941333333
10 18 2.48 0 0.5338738 0
11 20 2.23 4 0.9617071 -1.961333333
12 25 1.56 49 2.7247004 -11.55466667
13 30 0.94 144 5.1559271 -27.248
14 45 0.18 729 9.1849404 -81.828
15 60 0 1764 10.30838 -134.848
sum 270 48.16 3830 58.311293 -433.48
mean 18 3.210666667 sxx syy sxy
slope=b1=sxy/sxx -0.1132
intercept=b0=ybar-(slope*xbar) 5.2479
SST SYY 58.3113
SSR sxy^2/sxx 49.0613
SSE syy-sxy^2/sxx 9.2500
r^2 SSR/SST 0.8414
r sxy/sqrt(sxx*syy) -0.9173
error variance s^2 SSE/(n-2) 0.7115
S^2b1 s^2/sxx 0.0002
standard error b1=se(b1) sqrt(s^2b1) 0.0136
test statistics b1/se(b1) -8.3037
tcritical=For (1-alpha)% CI value of t 0.98 2.6245
Prediction at xi 56 -1.0902
standard error confidence interval for mean at prediction s*sqrt(1/n+ (xi-Xbar)^2/Sxx) 0.5619
Lower confidence interval for mean at given point xi y_pred -tcritical*s*sqrt(1/n+ (xi-Xbar)^2/Sxx) -2.5793
Upperr confidence interval for mean at given point xi y_pred +tcritical*s*sqrt(1/n+ (xi-Xbar)^2/Sxx) 0.3990
standard error Prediction interval for mean at given point xi s*sqrt(1+1/n+ (xi-Xbar)^2/Sxx) 1.0135
Lower Prediction interval for mean at given point xi y_pred -tcritical*s*sqrt(1+1/n+ (xi-Xbar)^2/Sxx) -2.2872
Upperr Prediction interval for mean at given point xi y_pred +tcritical*s*sqrt(1+1/n+ (xi-Xbar)^2/Sxx) 3.0851

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