Question

A dentist is interested in obtaining information about delinquent (past due) accounts. Since the practice opened 7 years ago, delinquent accounts are normally distributed with an average of 43 days and a variance of 121.00 days. The dentist randomly selected a sample of 23 delinquent accounts with an average of 47 days.

a) The dentist wants to know what average number of days is more than 38% from the past 7 years.


b) Is the average days from the sample of 23 more than the 38% average from the past 7 years?
---Select--- na Yes No

Answer 1

SOLUTION:

From given data,

A dentist is interested in obtaining information about delinquent (past due) accounts. Since the practice opened 7 years ago, delinquent accounts are normally distributed with an average of 43 days and a variance of 121.00 days. The dentist randomly selected a sample of 23 delinquent accounts with an average of 47 days.

Where,

Sample size = n = 23

Mean = = 43

Variance = = 121

Standard deviation = = sqrt(121) = 11

The sampling distribution for the sample mean is,

Mean of = = 43

Standard deviation of = = /sqrt(n) =  11 /sqrt(23) = 2.2936

We know that,

Z = ( - ) /

(a) The dentist wants to know what average number of days is more than 38% from the past 7 years.

38% = 38/100 = 0.38

P (> a) = 0.38

P (Z  > a - 43 / 2.2936) = 0.38

1 - P (Z  < a - 43 / 2.2936) = 0.38

P (Z  < a - 43 / 2.2936) = 0.62 --------- (1)

P (Z  < - 0.305481) = 0.62 (from normal table) ----------- (2)

By comparing (1) & (2) then we get,

a - 43 / 2.2936 = - 0.305481

a - 43 = - 0.305481 *2.2936

a - 43 = -0.7006512216

a   = -0.7006512216+43

a   = 42.29

42 days is more than 38%

b) Is the average days from the sample of 23 more than the 38% average from the past 7 years?

a sample of 23 delinquent accounts with an average of 47 days.

That is

For 23 days   = 47 days

Number of days more than 38% is

= 42.29

Z = ( - ) / = ( 47 - 42.29 ) / 2.2936 = ​​​​​​​ 2.05354

Here,

Z - value is not with in -2 & 2 , So this is usual value.

Hence, sample of 23 is more than the 38% average from the past 7 years.

Answer : Yes