Question

1, The data shown to the right represent the age​ (in weeks) at which babies first​ crawl, based on a survey of 12 mothers. Complete parts​ (a) through​ (c) below. 52 30 44 35 47 37 56 26 30 35 26 30

​(c) Construct and interpret a 95​% confidence interval for the mean age at which a baby first crawls. Select the correct choice and fill in the answer boxes to complete your choice.

​(Round to one decimal place as​ needed.)

A.The lower bound is weeks and the upper bound is weeks. We are 95​% confident that the mean age at which a baby first crawls is within the confidence interval.

B.The lower bound is weeks and the upper bound is weeks. We are 95​% confident that the mean age at which a baby first crawls is outside of the confidence interval.

2, The accompanying data represent the total travel tax​ (in dollars) for a​ 3-day business trip in 8 randomly selected cities. A normal probability plot suggests the data could come from a population that is normally distributed. A boxplot indicates there are no outliers. Complete parts​ (a) through​ (c) below.

68.94

79.32

69.95

84.51

80.37

85.93

101.91

99.35

b) Construct and interpret a 95​% confidence interval for the mean tax paid for a​ three-day business trip.

Select the correct choice below and fill in the answer boxes to complete your choice.

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

A.There is a ​% probability that the mean travel tax for all cities is between ​$ and ​$ .

B.One can be % confident that the mean travel tax for all cities is between ​$ and  ​$ .

C.One can be % confident that the all cities have a travel tax between ​$ and ​$ .

D.The travel tax is between $   and ​$ for % of all cities.

​(c) What would you recommend to a researcher who wants to increase the precision of the​ interval, but does not have access to additional​ data?

Answer 1

1)

sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   10.1384
Sample Size ,   n =    12
Sample Mean,    x̅ = ΣX/n =    37.3333

Level of Significance ,    α =    0.05          
degree of freedom=   DF=n-1=   11          
't value='   tα/2=   2.201   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   10.1384   / √   12   =   2.9267
margin of error , E=t*SE =   2.2010   *   2.9267   =   6.4417
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    37.33   -   6.441655   =   30.8917
Interval Upper Limit = x̅ + E =    37.33   -   6.441655   =   43.7750
95%   confidence interval is (   30.9 < µ <   43.8 )

A.The lower bound is 30.9 weeks and the upper bound is 43.8 weeks. We are 95​% confident that the mean age at which a baby first crawls is within the confidence interval.

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2)

sample std dev ,    s = √(Σ(X- x̅ )²/(n-1) ) =   12.0670
Sample Size ,   n =    8
Sample Mean,    x̅ = ΣX/n =    83.7850

Level of Significance ,    α =    0.05          
degree of freedom=   DF=n-1=   7          
't value='   tα/2=   2.365   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   12.0670   / √   8   =   4.2663
margin of error , E=t*SE =   2.3646   *   4.2663   =   10.0883
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    83.79   -   10.088283   =   73.6967
Interval Upper Limit = x̅ + E =    83.79   -   10.088283   =   93.8733
95%   confidence interval is (   73.70   < µ <   93.87   )

B.One can be 95 % confident that the mean travel tax for all cities is between ​$73.70 and  ​$93.87

c)

The researcher could decrease the level of confidence.