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Katarina [22]
2 years ago
14

Rockwell hardness of pins of a certain type is known to have a mean value of 50 and a standard deviation of 1.2.a. If the distri

bution is normal, what is the probability that the sample mean hardness for a random sample of 9 pins is at least 51?b. Without assuming population normality, what is the (approximate) probability that the sample mean hardness for a random sample of 40 pins is at least 51?
Mathematics
1 answer:
zalisa [80]2 years ago
5 0

Answer:

a

 P(\= X \ge 51 ) =0.0062

b

P(\= X \ge 51 ) = 0

Step-by-step explanation:

From the question we are told that

The mean value is \mu = 50

The standard deviation is  \sigma = 1.2

Considering question a

The sample size is  n = 9

Generally the standard error of the mean is mathematically represented as

      \sigma_x = \frac{\sigma }{\sqrt{n} }

=>   \sigma_x = \frac{ 1.2 }{\sqrt{9} }

=>  \sigma_x = 0.4

Generally the probability that the sample mean hardness for a random sample of 9 pins is at least 51 is mathematically represented as

      P(\= X \ge 51 ) = P( \frac{\= X - \mu }{\sigma_{x}}  \ge \frac{51 - 50 }{0.4 } )

\frac{\= X -\mu}{\sigma }  =  Z (The  \ standardized \  value\  of  \ \= X )

     P(\= X \ge 51 ) = P( Z  \ge 2.5 )

=>   P(\= X \ge 51 ) =1-  P( Z  < 2.5 )

From the z table  the area under the normal curve to the left corresponding to  2.5  is

    P( Z  < 2.5 ) = 0.99379

=> P(\= X \ge 51 ) =1-0.99379

=> P(\= X \ge 51 ) =0.0062

Considering question b

The sample size is  n = 40

   Generally the standard error of the mean is mathematically represented as

      \sigma_x = \frac{\sigma }{\sqrt{n} }

=>   \sigma_x = \frac{ 1.2 }{\sqrt{40} }

=>  \sigma_x = 0.1897

Generally the (approximate) probability that the sample mean hardness for a random sample of 40 pins is at least 51 is mathematically represented as  

       P(\= X \ge 51 ) = P( \frac{\= X - \mu }{\sigma_x}  \ge \frac{51 - 50 }{0.1897 } )

=> P(\= X \ge 51 ) = P(Z  \ge 5.2715  )

=>  P(\= X \ge 51 ) = 1- P(Z < 5.2715  )

From the z table  the area under the normal curve to the left corresponding to  5.2715 and

=>  P(Z < 5.2715  ) = 1

So

   P(\= X \ge 51 ) = 1- 1

=> P(\= X \ge 51 ) = 0

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