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dem82 [27]
3 years ago
7

The diameters of bolts produced in a machine shop are normally distributed with a mean of 5.725.72 millimeters and a standard de

viation of 0.070.07 millimeters. Find the two diameters that separate the top 5%5% and the bottom 5%5%. These diameters could serve as limits used to identify which bolts should be rejected. Round your answer to the nearest hundredth, if necessary.
Mathematics
2 answers:
suter [353]3 years ago
7 0

Answer:

Top 5% is 5.84 milliters and the bottom 5% is 5.60 millimeters.

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the zscore of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

\mu = 5.72, \sigma = 0.07

Top 5%:

X when Z has a pvalue of 0.95. So X when Z = 1.645

Z = \frac{X - \mu}{\sigma}

1.645 = \frac{X - 5.72}{0.07}

X - 5.72 = 1.645*0.07

X = 5.84

Bottom 5%:

X when Z has a pvalue of 0.05. So X when Z = -1.645

Z = \frac{X - \mu}{\sigma}

-1.645 = \frac{X - 5.72}{0.07}

X - 5.72 = -1.645*0.07

X = 5.60

Top 5% is 5.84 milliters and the bottom 5% is 5.60 millimeters.

lozanna [386]3 years ago
4 0

Answer:

The two diameters that separate the top 5% and the bottom 5% are 5.84 and 5.60 respectively.

Step-by-step explanation:

We are given that the diameters of bolts produced in a machine shop are normally distributed with a mean of 5.72 millimeters and a standard deviation of 0.07 millimeters.

<em>Let X = diameters of bolts produced in a machine shop</em>

So, X ~ N(\mu=5.72,\sigma^{2} = 0.07^{2})

The z score probability distribution is given by;

         Z = \frac{X-\mu}{\sigma} ~ N(0,1)

where, \mu = mean diameter = 5.72 millimeter

            \sigma = standard deviation = 0.07 millimeter

<u>Now, we have to find the two diameters that separate the top 5% and the bottom 5%.</u>

  • Firstly, Probability that the diameter separate the top 5% is given by;

       P(X > x) = 0.05

       P( \frac{X-\mu}{\sigma} > \frac{x-5.72}{0.07} ) = 0.05

       P(Z > \frac{x-5.72}{0.07} ) = 0.05

<em>So, the critical value of x in z table which separate the top 5% is given as 1.6449, which means;</em>

                 \frac{x-5.72}{0.07}  = 1.6449

               {x-5.72} = 1.6449 \times {0.07}

                        x   = 5.72 + 0.115143 = <u>5.84</u>

  • Secondly, Probability that the diameter separate the bottom 5% is given by;

       P(X < x) = 0.05

       P( \frac{X-\mu}{\sigma} < \frac{x-5.72}{0.07} ) = 0.05

       P(Z < \frac{x-5.72}{0.07} ) = 0.05

<em>So, the critical value of x in z table which separate the bottom 5% is given as -1.6449, which means;</em>

                 \frac{x-5.72}{0.07}  = -1.6449

               {x-5.72} = -1.6449 \times {0.07}

                        x   = 5.72 - 0.115143 = <u>5.60</u>

Therefore, the two diameters that separate the top 5% and the bottom 5% are 5.84 and 5.60 respectively.

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