The PGA would like to test to see if there is any difference in the distance produced by Titleist and Maxfli brand golf balls. A sample of 36 Titleist golf balls produced x1bar = 292ft, s1=16ft, while a sample of 48 Maxfli golf balls produced x2bar = 281ft, s2 20ft. Find a 90% confidence interval for the difference in mean distances.
To find the 90% confidence interval for the difference in mean distances between Titleist and Maxfli golf balls, we can use the two-sample t-test for independent samples.
First, let's calculate the standard error of the difference in means:
Standard Error (SE) = sqrt[ (s1^2 / n1) + (s2^2 / n2) ]
Where:
s1 = standard deviation of sample 1
n1 = sample size of sample 1
s2 = standard deviation of sample 2
n2 = sample size of sample 2
Plugging in the given values:
SE = sqrt[ (16^2 / 36) + (20^2 / 48) ]
≈ sqrt[ 7.11 + 8.33 ]
≈ sqrt[ 15.44 ]
≈ 3.93
Next, let's compute the degrees of freedom (df) using the formula:
df = ( (s1^2 / n1 + s2^2 / n2)^2 ) / ( ((s1^2 / n1)^2) / (n1 - 1) + ((s2^2 / n2)^2) / (n2 - 1) )
Plugging in the given values:
df = ( (16^2 / 36 + 20^2 / 48)^2 ) / ( ((16^2 / 36)^2) / (36 - 1) + ((20^2 / 48)^2) / (48 - 1) )
≈ ( (7.11 + 8.33)^2 ) / ( ((7.11)^2) / 35 + ((8.33)^2) / 47 )
≈ ( 15.44^2 ) / ( 1.42 + 1.09 )
≈ ( 238.27 ) / ( 2.51 )
≈ 94.82
Now, we can calculate the margin of error (ME) using the t-distribution critical value associated with a 90% confidence level and the calculated standard error:
ME = t * SE
The t-critical value can be found using a t-table or a calculator. Since we have a large enough sample (both 36 and 48 are greater than 30), we can use the z-value instead. For a 90% confidence level, the z-value is approximately 1.645.
ME ≈ 1.645 * 3.93
≈ 6.46
Finally, we can calculate the confidence interval by subtracting and adding the margin of error to the difference in means:
CI = (x1bar - x2bar) ± ME
CI = (292 - 281) ± 6.46
= 11 ± 6.46
Therefore, the 90% confidence interval for the difference in mean distances between Titleist and Maxfli golf balls is approximately 4.54 to 17.46 feet.