Let x be a random variable that represents the percentage of successful free throws a professional basketball player makes in a season. Let y be a random variable that represents the percentage of successful field goals a professional basketball player makes in a season. A random sample of n = 6 professional basketball players gave the following information.   x 69 66 63 65 76 76 y 54 53 51 43 47 45 ​ Verify that Se ≈ 4.739, a ≈ 67.050, b ≈ –0.263, and , ∑x =415, ∑y =293, ∑x2 =28,863, and ∑y2 =14,409, and find a 90% confidence interval for β and interpret its meaning. Round your final answers to three decimal places. answer choices: The 90% confidence interval for β is from –0.993 to 0.466 and means that for every percentage increase in successful free throws, there is 90% confidence that the percentage of successful field goals increases by an amount between –0.99 and 0.47.   The 90% confidence interval for β is from –1.065 to 0.539 and means that for every percentage increase in successful free throws, there is 90% confidence that the percentage of successful field goals increases by an amount between –1.07 and 0.54.   The 90% confidence interval for β is from –0.882 to 0.355 and means that for every percentage increase in successful free throws, there is 90% confidence that the percentage of successful field goals increases by an amount between –0.88 and 0.36.   The 90% confidence interval for β is from –1.023 to 0.496 and means that for every percentage increase in successful free throws, there is 90% confidence that the percentage of successful field goals increases by an amount between –1.02 and 0.50.   The 90% confidence interval for β is from –1.309 to 0.782 and means that for every percentage increase in successful free throws, there is 90% confidence that the percentage of successful field goals increases by an amount between –1.31 and 0.78.

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Let x be a random variable that represents the percentage of successful free throws a professional basketball player makes in a season. Let y be a random variable that represents the percentage of successful field goals a professional basketball player makes in a season. A random sample of n = 6 professional basketball players gave the following information.
 

x 69 66 63 65 76 76
y 54 53 51 43 47 45

Verify that Se ≈ 4.739, a ≈ 67.050, b  –0.263, and , ∑x =415, ∑y =293, ∑x2 =28,863, and ∑y2 =14,409, and find a 90% confidence interval for β and interpret its meaning. Round your final answers to three decimal places.

answer choices:
The 90% confidence interval for β is from –0.993 to 0.466 and means that for every percentage increase in successful free throws, there is 90% confidence that the percentage of successful field goals increases by an amount between –0.99 and 0.47.
 
The 90% confidence interval for β is from –1.065 to 0.539 and means that for every percentage increase in successful free throws, there is 90% confidence that the percentage of successful field goals increases by an amount between –1.07 and 0.54.
 
The 90% confidence interval for β is from –0.882 to 0.355 and means that for every percentage increase in successful free throws, there is 90% confidence that the percentage of successful field goals increases by an amount between –0.88 and 0.36.
 
The 90% confidence interval for β is from –1.023 to 0.496 and means that for every percentage increase in successful free throws, there is 90% confidence that the percentage of successful field goals increases by an amount between –1.02 and 0.50.
 
The 90% confidence interval for β is from –1.309 to 0.782 and means that for every percentage increase in successful free throws, there is 90% confidence that the percentage of successful field goals increases by an amount between –1.31 and 0.78.
Expert Solution
Step 1

Let x be a random variable that represents the percentage of successful free throws a professional basketball player makes in a season

Let y be a random variable that represents the percentage of successful field goals a professional basketball player makes in a season. 

n=6

The excel steps for regression is obtained as-

The steps taken to obtain the simple linear regression to determine the amount of time spent on homework can be predicted by amount of sleep is given as follows: -

  • Enter the data in Excel spreadsheet [ x and y]
  • Go to data>Go to data analysis>Select regression>click OK.
  • Select Input Y Range>Select Input X Range>click Labels>click Ok

The output obtained is given as follows-

Regression Statistics          
Multiple R 0.331          
R Square 0.109          
Adjusted R Square -0.113          
Standard Error 4.739          
Observations 6          
             
ANOVA            
  df SS MS F Significance F  
Regression 1 11.01801329 11.018013 0.490696 0.522220859  
Residual 4 89.81532004 22.45383      
Total 5 100.8333333        
             
  Coefficients Standard Error t Stat P-value Lower 90.0% Upper 90.0%
Intercept 67.050 26.078 2.571 0.062 11.457 122.644
x -0.263 0.376 -0.700 0.522 -1.065 0.539

The table below shows the required calculation-

  x y x2 y2
  69 54 4761 2916
  66 53 4356 2809
  63 51 3969 2601
  65 43 4225 1849
  76 47 5776 2209
  76 45 5776 2025
Total 415 293 28863 14409

Thus,

se4.739a67.050b-0.263ix=415jy=293ix2=28863jy2=14409

is verified.

 

 

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