$$\begin{array} { | l | l | }
\hline \text { Length (inch) } & \text { Frequency } \
\hline 0.1 & 2 \
\hline 0.2 & 3 \
\hline 0.3 & 3 \
\hline 0.4 & 5 \
\hline 0.5 & 4 \
\hline 0.6 & 2 \
\hline 0.7 & 2 \
\hline
\end{array}$$ Adele is studying a new species of insect that emits an odor perceived to be pleasant to humans. The known population consists of 21 insects. Adele has measured the length of each one.
Adele discovers a new specimen of this insect species. It is 0.5 inches long. What is its z-score?
A) 0.50
B) 0.61
C) 0.71
D) 0.80

Question

$$\begin{array} { | l | l | } 
\hline \text { Length (inch) } & \text { Frequency } \
\hline 0.1 & 2 \
\hline 0.2 & 3 \
\hline 0.3 & 3 \
\hline 0.4 & 5 \
\hline 0.5 & 4 \
\hline 0.6 & 2 \
\hline 0.7 & 2 \
\hline
\end{array}$$ Adele is studying a new species of insect that emits an odor perceived to be pleasant to humans. The known population consists of 21 insects. Adele has measured the length of each one.
Adele discovers a new specimen of this insect species. It is 0.5 inches long. What is its z-score?
A) 0.50
B) 0.61
C) 0.71
D) 0.80

Quizplus · Accepted Answer

To find the z-score of the new specimen which is 0.5 inches long, we need to use the formula:
z = (x - μ) / σ

where x is the length of the new specimen, μ is the mean length of the population, and σ is the standard deviation of the population.

First, we need to calculate the mean and standard deviation of the population using the given frequency distribution. We can use the formula:

μ = Σ(x * f) / Σf

where Σ is the sum over all values, x is the length of the insect, and f is the frequency of that length.

We have:
μ = (0.1*2 + 0.2*3 + 0.3*3 + 0.4*5 + 0.5*4 + 0.6*2 + 0.7*2) / 21 ≈ 0.39

Next, we need to find the standard deviation of the population, which can be calculated using the formula:

σ = sqrt[Σ(f * (x - μ)^2) / Σf]

We have:
σ = sqrt[(2*(0.1-0.39)^2 + 3*(0.2-0.39)^2 + 3*(0.3-0.39)^2 + 5*(0.4-0.39)^2 + 4*(0.5-0.39)^2 + 2*(0.6-0.39)^2 + 2*(0.7-0.39)^2) / 21] ≈ 0.14

Now, we can substitute these values into the formula for z-score:
z = (0.5 - 0.39) / 0.14 ≈ 0.61

Therefore, the z-score of the new specimen is 0.61, which corresponds to choice (B).

Adele Is Studying a New Species of Insect That Emits