Question

Sharon is considering trying to open a new business within the next few years, and she is doing research to determine what kind of businesses tend to be successful. So far, she has compiled two tables. The first shows the number of businesses of several types started in Sharon’s city over the course of two years, and the number of those businesses which did not succeed and were forced to shut down within two years of opening. The second deals with separate records of successful new businesses, showing how much profit those new businesses turned over two years. Businesses on the boundary lines fall in the lower category.
Type
Food
Retail
Financial
Service
Opened
3,193
2,280
1,898
5,045
Closed
1,977
1,626
1,443
3,548


Up to $25k
$25-50k
$50-75k
$75-100k
Over $100k
Food
945
623
601
258
114
Retail
813
548
347
188
63
Financial
316
244
195
86
51
Service
979
739
432
174
124

Using the tables as experimental data, determine whether it is more likely for a new retail establishment to succeed and earn profits of $50,000 or more, or whether it is more likely for a new service establishment to succeed and earn profits of $25,000 or less, and how much more likely the one situation is than the other. Express all probabilities as percentages to two decimal places, and express differences by number of percentage points (for example, 23% is 5 percentage points greater than 18%).
a.
The situation involving the retail establishment has a probability 0.99 percentage points higher than the situation involving the service establishment.
b.
The situation involving the retail establishment has a probability 4.91 percentage points higher than the situation involving the service establishment.
c.
The situation involving the service establishment has a probability 5.66 percentage points higher than the situation involving the retail establishment.
d.
The situation involving the service establishment has a probability 6.35 percentage points higher than the situation involving the retail establishment.

Answer 1

Compute the two probabilities from the tables.

Retail: success rate = (opened − closed)/opened = (2280 − 1626)/2280 = 654/2280.
Among successful retail, fraction with ≥$50k = (347+188+63)/1959 = 598/1959.
So P(retail succeeds and earns ≥$50k) = (654/2280)·(598/1959) = 0.08761 = 8.76%.

Service: success rate = (5045 − 3548)/5045 = 1497/5045.
Among successful service, fraction with ≤$25k = 979/2448.
So P(service succeeds and earns ≤$25k) = (1497/5045)·(979/2448) = 0.11871 = 11.87%.

Comparison: the service case is more likely. Difference = 11.87% − 8.76% = 3.11 percentage points.

None of the provided choices (a–d) match this result.