Question

Dottie's Tax Service specializes in federal tax returns for professional clients, such as physicians, dentists, accountants, and lawyers. A recent audit by the IRS of the returns she prepared indicated that an error was made on 11% of the returns she prepared last year. Assuming this rate continues into this year and she prepares 67 returns, what is the probability that she makes errors on:

a. More than 7 returns? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places.) Probability

b. At least 7 returns? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places.) Probability

c. Exactly 7 returns? (Round z-score computation to 2 decimal places and your final answer to 4 decimal places.) Probability

Answer 1

Answer :

Given data is :

Probability of tax return p = 11% = 0.11

then q = 1 - p = 1 - 0.11 = 0.89

and No.of tax returns (Sample size) n = 67

therefore,

Mean = = 67 * 0.11 = 7.37

Mean = 7.37

Standard deviation =

=

=

=

Standard deviation = 2.56

a)Here we need to find the probability that she makes errors on more than 7 returns is : X + 0.5

i.e probability that more than X occurs.

i.e 7 + 0.5 = 7.5

therefore,

Z value with = 7.5

Z = (7.5 - 7.37) / 2.56

= 0.13 / 2.56

= 0.051

Z = 0.05

Now,consider

P(7.37 < < 7.5) =

=

=

=

= P(Z < 0.05) - P(Z < 0)

= 0.5199 - 0.5000

= 0.0199

P(7.37 < < 7.5) = 0.0199

therefore,

P( > 7.5) = 0.5 - P(7.37 < < 7.5)

= 0.5 - 0.0199

= 0.4801

P( > 7.5) = 0.4801

b)Here we need to find the probability that X is 7 or more.

i.e X - 0.5 = 7 - 0.5 = 6.5

i.e = 6.5

Z = (6.5 - 7.37) / 2.56

= -0.87 / 2.56

= -0.34

Z = -0.34

Now,consider

P(6.5 < < 7.37) =

=

=

=

= P(Z < 0) - P(Z < -0.34)

= 0.5000 - 0.3669

= 0.1331

P(6.5 < < 7.37) = 0.1331

Now,

P( > 6.5) = 0.5 + P(6.5 < < 7.37)

= 0.5 + 0.1331

= 0.6331

P( > 6.5) = 0.6331

c)Here we need to find that probability X is exactly 7 :

i.e P(X = 7)

P(X = 7) = P(X is 7 or more) - P(X is more than 7)

= P( > 6.5) - P( > 7.5)

= 0.6331 - 0.4801

= 0.1530

P(X = 7) = 0.1530