Question

Century National Bank has offices in several cities in the Midwest and the southeastern part of the United States. Mr. Dan Selig, president and CEO would like to know the characteristics of his checking account customers. When Mr. Selig took over as president of Century National Bank, several years ago, the use of debit cards was just beginning. He would like an update on the use of these cards. Develop a 95% confidence interval for the proportion of customers who have debit cards. On the basis of the confidence interval, is it reasonable to conclude that less than half the customers use a debit card? Write a brief report to Mr. Selig interpreting your results.

70 Accounts | |||||

Balance | ATM | Services | Debit | Interest | City |

$1,756.00 | 13 | 4 | 0 | 1 | 2 |

$748.00 | 9 | 2 | 1 | 0 | 1 |

$1,501.00 | 10 | 1 | 0 | 0 | 1 |

$1,831.00 | 10 | 4 | 1 | 1 | 3 |

$1,622.00 | 14 | 6 | 1 | 1 | 4 |

$1,886.00 | 17 | 3 | 0 | 1 | 1 |

$740.00 | 6 | 3 | 0 | 0 | 3 |

$1,593.00 | 10 | 8 | 1 | 0 | 1 |

$1,169.00 | 6 | 4 | 0 | 0 | 4 |

$2,125.00 | 18 | 6 | 1 | 0 | 2 |

$1,554.00 | 12 | 6 | 1 | 0 | 3 |

$1,474.00 | 12 | 7 | 1 | 0 | 1 |

$1,913.00 | 6 | 5 | 0 | 0 | 1 |

$1,218.00 | 10 | 3 | 1 | 0 | 1 |

$1,006.00 | 12 | 4 | 0 | 0 | 1 |

$2,215.00 | 20 | 3 | 1 | 0 | 4 |

$137.00 | 7 | 2 | 0 | 0 | 3 |

$167.00 | 5 | 4 | 1 | 0 | 4 |

$343.00 | 7 | 2 | 1 | 0 | 1 |

$2,557.00 | 20 | 7 | 1 | 0 | 4 |

$2,276.00 | 15 | 4 | 1 | 0 | 3 |

$1,494.00 | 11 | 2 | 1 | 1 | 1 |

$2,144.00 | 17 | 3 | 0 | 0 | 3 |

$1,995.00 | 10 | 7 | 0 | 0 | 2 |

$1,053.00 | 8 | 4 | 1 | 0 | 3 |

$1,526.00 | 8 | 4 | 1 | 1 | 2 |

$1,120.00 | 8 | 6 | 1 | 0 | 3 |

$1,838.00 | 7 | 5 | 0 | 1 | 3 |

$1,746.00 | 11 | 2 | 0 | 0 | 2 |

$1,616.00 | 10 | 4 | 1 | 1 | 2 |

$1,958.00 | 6 | 2 | 0 | 0 | 2 |

$634.00 | 2 | 7 | 0 | 1 | 4 |

$580.00 | 4 | 1 | 1 | 0 | 1 |

$1,320.00 | 4 | 5 | 1 | 0 | 1 |

$1,675.00 | 6 | 7 | 1 | 0 | 2 |

$789.00 | 8 | 4 | 0 | 0 | 4 |

$1,735.00 | 12 | 7 | 1 | 1 | 3 |

$1,784.00 | 11 | 5 | 0 | 0 | 1 |

$1,326.00 | 16 | 8 | 1 | 0 | 3 |

$2,051.00 | 14 | 4 | 0 | 0 | 4 |

$1,044.00 | 7 | 5 | 1 | 0 | 1 |

$1,885.00 | 10 | 6 | 1 | 1 | 2 |

$1,790.00 | 11 | 4 | 1 | 1 | 3 |

$765.00 | 4 | 3 | 0 | 0 | 4 |

$1,645.00 | 6 | 9 | 0 | 1 | 4 |

$32.00 | 2 | 0 | 1 | 0 | 3 |

$1,266.00 | 11 | 7 | 0 | 0 | 4 |

$890.00 | 7 | 1 | 0 | 1 | 1 |

$2,204.00 | 14 | 5 | 1 | 0 | 2 |

$2,409.00 | 16 | 8 | 1 | 0 | 2 |

$1,338.00 | 14 | 4 | 1 | 0 | 2 |

$2,076.00 | 12 | 5 | 0 | 0 | 2 |

$1,708.00 | 13 | 3 | 1 | 0 | 1 |

$2,138.00 | 18 | 5 | 1 | 1 | 4 |

$2,375.00 | 12 | 4 | 1 | 0 | 2 |

$1,455.00 | 9 | 5 | 1 | 1 | 3 |

$1,487.00 | 8 | 4 | 0 | 0 | 4 |

$1,125.00 | 6 | 4 | 0 | 0 | 2 |

$1,989.00 | 12 | 3 | 1 | 1 | 2 |

$2,156.00 | 14 | 5 | 1 | 0 | 2 |

$1,735.00 | 12 | 7 | 1 | 1 | 3 |

$1,766.00 | 11 | 5 | 0 | 0 | 1 |

$3,326.00 | 16 | 8 | 1 | 0 | 3 |

$4,051.00 | 14 | 4 | 1 | 0 | 4 |

$1,974.00 | 7 | 5 | 1 | 1 | 1 |

$2,735.00 | 18 | 6 | 0 | 0 | 2 |

$1,664.00 | 12 | 6 | 1 | 1 | 3 |

$1,874.00 | 12 | 7 | 1 | 0 | 1 |

$2,334.00 | 12 | 4 | 1 | 0 | 2 |

$1,495.00 | 9 | 5 | 0 | 1 | 3 |

Answer #1

In the column Debit

1---those who have debit card

0---those who do not possess debit card

p^=x/n=43/43+27

=43/70

=0.614286

sample proportion of customers who have debit cards=0.614286

95% confidence interval for true proportion of customers who have debit cards=

p^-Zsqqrt(p^(1-p^)/n,p^+Zsqqrt(p^(1-p^)/n

0.614286-1.96*sqrt(0.614286*(1-0.614286)/70),0.614286+1.96*sqrt(0.614286*(1-0.614286)/70

0.5003,0.7283

lower kimit=0.5003

upper limit=0.7283

we are 95% confident that the true proportion of customers who have debit cards ies in between

0.5003 and 0.7283.

For the qustion:

On the basis of the confidence interval, is it reasonable to conclude that less than half the customers use a debit card?

H0:p=0.50

H1:p<0.50

95% confident that the true proportion of customers who have debit cards ies in between

0.5003 and 0.7283.

Null hypothesis value p=0.50 lies in between interval.

0.5003<0.50<0.7283

Do not reject H0.

Conclsion:

there is no sufficient statistical evidence at 5% level of significance to conclude that less than half the customers use a debit card

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