Question

The Bank of Connecticut issues Visa and Mastercard credit cards. The balances on these credit cards follow a normal distribution with a mean of $845 and standard deviation of $270. What percentage of balances is between $600 and $1,000?

Answer 1

Answer:

53.37% of balances are between $600 and $1000

Step-by-step explanation:

Lets call X the balance of a cradit card. X is a normal distribution with mean 845 and standard deviation of 270.

We can standardize X so that it has distribution equal to the standard normal distribution. For that consider Z = (X-845)/270. Z is a standard normally distributed random variable.

We want to calculate P( 600 < X < 1000). That is the same as calculate P( (600-845)/270 < Z < (1000-845)/270) = P(-0.907 < Z < 0.574)

Remember that for a value of k we have that P(Z < k) = Φ(k), where Φ denotes the cummulative distribution function of a standard normal random variable. The values of Φ can be found on the table i attached with this answer.  

P(-0.907 < Z < 0.574) = P(Z < 0.574) - P(Z < -0.907) = Φ(0.574)-Φ(-0.907)

We can read the table to obtain that Φ(0.574) = 0.7157. To calculate Φ(-0.907) we can use the simetry of the normal function (on the y axis) to conlcude that Φ(-0.907) = 1 - Φ(0.907) = 1 - 0.8186 = 0.1814.

Finally P( 600 < X < 1000) = Φ(0.574)-Φ(-0.907) = 0.7157-0.1814 = 0.5337.

A proportion of 0.5337 of the balances are between 600 and 1000, that is a 53.37%

I hope this helped you!

Download pdf