When a dermatologist suspects that skin may show signs of disease or cancer they will likely take a skin biopsy for analysis. This
process involves removing a small portion of the affected skin. In a 2017 study, the proportion of skin biopsies that came back as
melanoma was .
If the study included 116 biopsies, are the conditions met to use a normal distribution approximation?
and The conditions are met
Submit

Ask by Gray Greene. in the United States

Mar 15,2025

Question

When a dermatologist suspects that skin may show signs of disease or cancer they will likely take a skin biopsy for analysis. This
process involves removing a small portion of the affected skin. In a 2017 study, the proportion of skin biopsies that came back as
melanoma was .
If the study included 116 biopsies, are the conditions met to use a normal distribution approximation?
and The conditions are met
Submit

Ask by Gray Greene. in the United States

Mar 15,2025

UpStudy AI · Accepted Answer

To determine if the conditions are met to use a normal distribution approximation for the proportion of skin biopsies that came back as melanoma, we need to check the following conditions:

1. $$ n p $$ should be greater than or equal to 10.
2. $$ n (1 - p) $$ should also be greater than or equal to 10.

Where:
- $$ n $$ is the number of biopsies (116 in this case).
- $$ p $$ is the proportion of biopsies that came back as melanoma (0.014).

Let's calculate $$ n p $$ and $$ n (1 - p) $$:

1. Calculate $$ n p $$:
   $$
   n p = 116 \times 0.014
   $$

2. Calculate $$ n (1 - p) $$:
   $$
   n (1 - p) = 116 \times (1 - 0.014)
   $$

Now, I will perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$116\times 0.014$$
- step1: Multiply the numbers:
  $$1.624$$
Calculate or simplify the expression $$ 116 * (1 - 0.014) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$116\left(1-0.014\right)$$
- step1: Subtract the numbers:
  $$116\times 0.986$$
- step2: Multiply the numbers:
  $$114.376$$
Now that we have performed the calculations, we can summarize the results:

1. $$ n p = 116 \times 0.014 = 1.624 $$
2. $$ n (1 - p) = 116 \times (1 - 0.014) = 114.376 $$

Next, we check the conditions:

- $$ n p = 1.624 $$ (which is less than 10)
- $$ n (1 - p) = 114.376 $$ (which is greater than 10)

Since $$ n p < 10 $$, the conditions are **not met** to use a normal distribution approximation.

Thus, the conclusion is:
- The conditions are **not met**.
The conditions are not met to use a normal distribution approximation because $$ n p = 1.624 $$ is less than 10.

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