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Bas_tet [7]
3 years ago
11

In a chemical process, the amount of a certain type of impurity in the output is difficult to control and is thus a random varia

ble. It is known that the standard deviation of the impurity is 0.0015 following an approximately normal distribution. If a random sample of 75 outputs has an average impurity of 0.310, find a 95% confidence interval for the mean amount of the impurity.
Chemistry
2 answers:
Julli [10]3 years ago
8 0

Answer:

+/- 0.00033

Explanation:

For a 95 % confidence interval the range is given by

+/- Z * s/sqrt(n)

where Z value is 1.916 for a 95% confidence interval

Therefore the interval is

s is the standard deviation in this case 0.0015

n is 75 the number of outposts

calculating,

+/- 1.916 * 0.0015/sqrt(75) = +/- 0.00033

strojnjashka [21]3 years ago
5 0

Answer:

\\ (Lower\;value =0.3097 ; Upper\;value = 0.3103).

We can also represent it as \\ P(0.3097 \leq \mu \leq 0.3103) = 0.95.

Explanation:

This is a case of finding <em>two values</em> that define an <em>interval</em> in which is included the <em>population mean for the impurity</em> in the chemical process with a probability of 95%.

We have full information to this respect to solve the question:

  1. The <em>population standard deviation</em>, which is \\ \sigma = 0.0015.
  2. The <em>sample mean</em>, which is \\ \bar{x} = 0.310.
  3. The <em>sample size</em> \\ n = 75.

That is, we can find the 95% confidence interval for a <em>given population standard deviation</em>.

The formula for finding these <em>two values, </em>in these conditions,<em> </em>is as follows:

\\ Lower\;value = \bar{x} - 1.96\frac{\sigma}{\sqrt{n}}.

\\ Upper\;value = \bar{x} + 1.96\frac{\sigma}{\sqrt{n}}.

We have already confirmed that we have all these values: the population standard deviation, the sample mean and the sample size.

The number 1.96 corresponds to a <em>z-score</em> that is<em> 1.96</em> times the standard deviation from the mean, and represents the <em>confidence coefficient</em> and depends on the <em>confidence level</em>, which is in this case of 5% (or 0.05). A confidence level of 5% determines this 95 % confidence interval.

Notice that we are going to use a mean obtained from a sample of 75 elements \\ \bar{x} = 0.310, which represents a <em>sample mean</em>.

As a result, in order to find the two values that define the <em>95% confidence interval, </em>we can proceed as follows:

Lower value of the 95% confidence interval

\\ Lower\;value = \bar{x} - 1.96\frac{\sigma}{\sqrt{n}}.

\\ Lower\;value = 0.310 - 1.96\frac{0.0015}{\sqrt{75}}.

\\ Lower\;value =0.3097.

Upper value of the 95% confidence interval

\\ Upper\;value = \bar{x} + 1.96\frac{\sigma}{\sqrt{n}}.

\\ Upper\;value = 0.310 + 1.96\frac{0.0015}{\sqrt{75}}.

\\ Upper\;value = 0.3103.

Thus, the 95% confidence interval, or the interval for which there is a probability of 95% to find the population mean for this certain type of impurity:

\\ Lower\;value =0.3097.

\\ Upper\;value = 0.3103.

That is:

\\ P(0.3097 \leq \mu \leq 0.3103) = 0.95.

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