Mathematics: The lengths of nails produced in a factory are normally distributed with a mean of 3.34 centimeters and a standard deviation of 0.07 centimeters. Find the two lengths that separate the top 3% and the bottom 3%. These lengths could serve as limits used to identify which nails should be rejected. Round your answer to the nearest hundredth, if necessary.

The lengths of nails produced in a factory are normally

QueensQueens

30.01.2022

3.47 and 3.21

Step-by-step explanation:

Let us assume the nails length be X

$X \sim N(3.34,0.07^2)$

Value let separated the top 3% is T and for bottom it would be B

P(X < T)= 0.97

Now converting, we get

$P(Z < \frac{T-3.34}{0.07})= 0.97$

Based on the normal standard tables, we get

P(Z < 1.881)= 0.97

Now compare these two above equations

$\frac{T-3.34}{0.07} = 1.881 \\\\ T = 1.881 \times 0.07 + 3.34 \\\\ = 3.47$

So for top 3% it is 3.47

Now for bottom we applied the same method as shown above

$P(Z < \frac{B-3.34}{0.07})= 0.03$

Based on the normal standard tables, we get

P(Z < -1.881)= 0.03

Now compare these two above equations

$\frac{B-3.34}{0.07} = -1.881$

$= -1.881 \times 0.07 + 3.34 \\\\ = 3.21$

hence, for bottom it would be 3.21

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Shu11

13.10.2022

$a_{n}$ = 4n + 2

Step-by-step explanation:

There is a common difference d between consecutive terms in the sequence, that is

d = 10 - 6 = 14 - 10 = 18 - 14 = 22 - 18 = 4

This indicates the sequence is arithmetic with n th term

$a_{n}$ = a₁ + (n - 1)d

where a₁ is the first term and d the common difference

Here a₁ = 6 and d = 4, thus

$a_{n}$ = 6 + 4(n - 1) = 6 + 4n - 4 = 4n + 2

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