Thompson and Thompson is a steel bolts manufacturing company. Their current steel bolts have a mean diameter of 125125 millimeters, and a va

Question

Thompson and Thompson is a steel bolts manufacturing company. Their current steel bolts have a mean diameter of 125125 millimeters, and a variance of 4949. If a random sample of 4141 steel bolts is selected, what is the probability that the sample mean would differ from the population mean by greater than 2.22.2 millimeters? Round your answer to four decimal places.

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Serenity 2 weeks 2021-09-11T02:56:42+00:00 1 Answer 0

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    2021-09-11T02:58:07+00:00

    Answer:

    The probability that the sample mean would differ from the population mean by greater than 2.2 millimeters is 0.0222.

    Step-by-step explanation:

    Let X = the diameter of steel bolts.

    The population mean diameter of steel bolts is, μ = 125 mm and the population variance is, σ² = 49 mm.

    A random sample of n = 41 steel bolts are selected.

    According to the Central limit theorem, if a large sample (n > 30) is selected from an unknown population with mean μ and variance σ² then the sampling distribution of sample mean follows a Normal distribution with mean, \mu_{\bar x}=\mu and variance, \sigma_{\bar x}^{2}=\frac{\sigma^{2}}{n}.

    Compute the probability that the sample mean differ from the population mean by greater than 2.2 mm as follows:

    P(\bar X-\mu>2.2)=P(\frac{\bar X-\mu}{\sigma_{\bar x}}>\frac{2.2}{\sqrt{49/41}})\\=P(Z>2.01)\\=1-P(Z<2.01)\\=1-0.9778\\=0.0222

    *Use a z-table for the probability.

    Thus, the probability that the sample mean would differ from the population mean by greater than 2.2 millimeters is 0.0222.

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