Question

A study of the ability of individuals to walk in a straight line reported the accompanying data on cadence (strides per second) for a sample of n = 20 randomly selected healthy men.

0.95 0.85 0.92 0.95 0.93 0.87 1.00 0.92 0.85 0.81
0.76 0.93 0.93 1.03 0.93 1.06 1.08 0.96 0.81 0.95

A normal probability plot gives substantial support to the assumption that the population distribution of cadence is approximately normal. A descriptive summary of the data from Minitab follows.

Variable	N	Mean	Median	TrMean	StDev	SEMean
cadence	20	0.9245	0.9300	0.9250	0.0827	0.0185
Variable		Min	Max	Q1	Q3
cadence		0.7600	1.0800	0.8600	0.9550

(a) Calculate and interpret a 95% confidence interval for population mean cadence. (Round your answers to four decimal places.)

(_,_) strides per second

Interpret this interval.

With 95% confidence, the value of the true mean cadence of all such men falls below the confidence interval.

With 95% confidence, the value of the true mean cadence of all such men falls inside the confidence interval.    

With 95% confidence, the value of the true mean cadence of all such men falls above the confidence interval.

(b) Calculate and interpret a 95% prediction interval for the cadence of a single individual randomly selected from this population. (Round your answers to four decimal places.)

(_,_) strides per second

Interpret this interval.

If this bound is calculated once, there is a 5% chance that these bounds will capture a future individual value of cadence for a healthy man.

If this bound is calculated once, there is a 95% chance that these bounds will capture a future individual value of cadence for a healthy man.    

If this bound is calculated sample after sample, in the long run, 95% of these bounds will fail to capture a future individual value of cadence for a healthy man.

If this bound is calculated sample after sample, in the long run, 95% of these bounds will capture a future individual value of cadence for a healthy man.

(c) Calculate an interval that includes at least 99% of the cadences in the population distribution using a confidence level of 95%. (Round your answers to four decimal places.)

(_,_) strides per second

Interpret this interval.

We can be 99% confident that the interval includes at least 95% of the cadence values in the population.

We can be 5% confident that the interval includes at least 99% of the cadence values in the population.    

We can be 95% confident that the interval includes at least 99% of the cadence values in the population.

We can be 1% confident that the interval includes at least 95% of the cadence values in the population.

Answer 1

a)

Level of Significance ,    α =    0.05
sample std dev ,    s =    0.0827
Sample Size ,   n =    20
Sample Mean,    x̅ =   0.9245
      
degree of freedom=   DF=n-1=   19
.

't value='   tα/2=   2.0930   [Excel formula =t.inv(α/2,df) ]  
              
Standard Error , SE =   s/√n =   0.0185      
margin of error ,   E=t*SE =   0.039

   
confidence interval is               
Interval Lower Limit=   x̅ - E =    0.8858      
Interval Upper Limit=   x̅ + E =    0.9632      
confidence interval is (   0.8858   < µ <   0.9632   )

With 95% confidence, the value of the true mean cadence of all such men falls inside the confidence interval.

b)

margin of error for prediction interval,E=   t*s*√(1+1/n)=   0.1775  

   
prediction interval is               
Interval Lower Limit=   x̅ - E =    0.7470      
Interval Upper Limit=   x̅ + E =    1.1020      
prediction interval is (   0.7470   < µ <   1.1020   )

If this bound is calculated once, there is a 95% chance that these bounds will capture a future individual value of cadence for a healthy man

c)

tolerance level =    x̅ ± K*s
tolerance critical value with n=20 and 99% confidence level is 3.615   

tolerance level =    x̅ ± K*s = 0.9245 ± 3.615*0.0827
(0.6254,1.2236)

We can be 95% confident that the interval includes at least 99% of the cadence values in the population