Question

Teacl Smolts are young salmon at a stage when their skin becomes covered with silvery scales and they start to migrate from freshwater to the sea. The reflectance of a light shined on a smolt's skin is a measure of the smolt's readiness for the migration. Here are the reflectances, in percents, for a sample of 50 smolts. 57.6 54.9 63.4 57.1 54.8 42.4 63.5 55.5 33.6 63.2 58.3 42.0 56.0 47.7 56.2 56.0 38.7 49.6 42.2 45.7 69.1 50.5 53.0 38.3 60.4 49.2 42.8 44.6 46.3 44.2 59.0 42.0 47.6 47.9 69.2 46.6 68.0 42.9 45.5 474 59.5 37.7 53.9 43.1 51.3 64.6 43.7 42.7 50.9 43.8 (a) Find the standard deviation of the reflectance for these smolts. (Round your answer to two decimal places.) (b) Find the quartiles of the reflectance for these smolts (c) Do you prefer the standard deviation or the quartiles as a measure of spread for these data? Give reasons for your preference.

Answer 1

Answer:

a) s = 8.81

b) Q1 = 43.4, Q2 = 49.4, Q3 = 57.35

c) See below

Step-by-step explanation:

(a) Find the standard deviation of the reflectance for these smolts. (Round your answer to two decimal places.)

In order to find the standard deviation, we need the mean first. The mean is defined as

$\bar x=\displaystyle\frac{\displaystyle\sum_{i=1}^{n}x_i}{n}$

where the $x_i$ are the values of the data collected and n=50 the size of the sample.

So, the mean is

$\bar x=50.882$

Now, the standard deviation of the sample is defined as  

$s=\sqrt{\displaystyle\frac{\displaystyle\sum_{i=1}^n(x_i-\bar x)^2}{n-1}}$

and we have that our standard deviation is

s = 8.81

(b) Find the quartiles of the reflectance for these smolts

To find the quartiles, we must sort the data from lowest to largest:  

33.6,  37.7,  38.3,  38.7,  42,  42,  42.2,  42.4,  42.7,  42.8,  42.9,  43.1,  43.7,  43.8,  44.2,  44.6,  45.5,  45.7,  46.3,  46.6,  47.4,  47.6,  47.7,  47.9,  49.2,  49.6,  50.5,  50.9,  51.3,  53,  53.9,  54.8,  54.9,  55.5,  56,  56,  56.2,  57.1,  57.6,  58.3,  59,  59.5,  60.4,  63.2,  63.4,  63.5,  64.6,  68,  69.1,  69.2

The first quartile is the number between the 12th and the 13th data (so 25% of the data are below it and 75% above it)

So the 1st quartile is

$Q_1=\displaystyle\frac{43.1+43.7}{2}=43.4$

The 2nd quartile is the median, the point between the 25th and 26th data, it splits the data in two halves.

$Q_2=\displaystyle\frac{49.2+49.6}{2}=49.4$

The 3rd quartile is the point between the 38th and 39th data (so 75% of the data are below it and 25% above it)

$Q_3=\displaystyle\frac{57.1+57.6}{2}=57.35$

(c) Do you prefer the standard deviation or the quartiles as a measure of spread for these data? Give reasons for your preference.

In this case, we prefer the quartiles as a measure of spread since the data are very scattered around the mean and there is no a central tendency.