6.3.97 z-score, percentage

According to a recent study, the carapace length for adult males of a certain species of tarantula are normally distributed with a mean of \(\mu\) = 17.14 mm and a standard deviation of \(\sigma\) = 1.88 mm. Complete parts​ (a) through​ (d) below.

(a)Find the percentage of the tarantulas that have a carapace length between 15 mm and 16 mm.

First, we need to find z-score for 15mm and 16mm by using the formula \(z=\frac{x-\mu}{\sigma}\)

(15-17.14)/1.88

## [1] -1.138298

(16-17.14)/1.88 

## [1] -0.606383

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We get each probability to the left by using pnorm() command or using the table

pnorm((16-17.14)/1.88 )

## [1] 0.2721302

pnorm((15-17.14)/1.88 )

## [1] 0.1274981

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In order to get the data between the range 15 and 16 we subtract these probabilities above

pnorm((16-17.14)/1.88)-pnorm((15-17.14)/1.88)

## [1] 0.1446322

Round the answer two four decimal places

round(pnorm((16-17.14)/1.88)-pnorm((15-17.14)/1.88), 4)

## [1] 0.1446

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(b) Find the percentage of the tarantulas that have a carapace length exceeding 18 mm.

First, we need to find z-score for 18mm

(18-17.14)/1.88

## [1] 0.4574468

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We get each probability to the left by using pnorm() command or using the table

pnorm((18-17.14)/1.88 )

## [1] 0.676325

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In order to get the data exceeding 18 mm

1-pnorm((18-17.14)/1.88)

## [1] 0.323675

Round the answer two four decimal places

round(1-pnorm((18-17.14)/1.88), 4)

## [1] 0.3237

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(c) Determine and interpret the quartiles for the carapace length of these tarantulas.

The area to the left of the first quartile is .25

Using qnorm() to get the z-value

qnorm(.25)

## [1] -0.6744898

Using the formula \(x = \mu + \sigma . z\)

17.14 + 1.88 * qnorm(.25)

## [1] 15.87196

round(17.14 + 1.88 * qnorm(.25), 2)

## [1] 15.87

Since the area to the left of second and third quartile are .5 and .75 Using the same approach we have

round(17.14 + 1.88 * qnorm(.5), 2)

## [1] 17.14

round(17.14 + 1.88 * qnorm(.75), 2)

## [1] 18.41

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(d) Obtain the 95th percentile for the carapace length of these tarantulas

The area to the left of 95th percentile is .95

round(17.14 + 1.88 * qnorm(.95), 2)

## [1] 20.23

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Hope that helps!