How To Find The Best Point Estimate Of The Population Mean

A point gauge of the mean of a population is determined past calculating the mean of a sample drawn from the population. The calculation of the hateful is the sum of all sample values divided by the number of values.

$$ \large\displaystyle \bar{X}=\frac{\sum\nolimits_{i=1}^{north}{{{x}_{i}}}}{n}$$

Where $- \bar{X} -$ is the hateful of the n individual ten_i values.

The larger the sample the more accurate the judge. If n equals the number of items in the population the same formula calculates the population mean, $- \mu -$.

Estimate of location of $- \mu -$

The same of n items besides can provide an interval that indicates where the population mean, $- \mu -$, resides with a stated probability or confidence. If nosotros say the unknown population hateful has a ninety% chance of residing inside an interval, we are using a 90% confidence interval. This as well ways in that location is a 10% chance $- \mu -$ falls outside the interval.

The formula for the confidence interval well-nigh an estimated hateful is dependent on whether or not we know the population standard deviation, $- \sigma -$. Generally we exercise not know $- \sigma -$ and utilize an estimated standard deviation, s, using the sample values. The sample's mean confidence interval is

$$ \large\displaystyle \bar{Ten}\pm {{t}_{\alpha /2,\text{ }n-ane}}\frac{s}{\sqrt{n}}$$

Where $- {{t}_{\alpha /two,\text{ }n-1}} -$ is the t-distribution with 1-$- \alpha -$ confidence and north-1 degrees of freedom value. s is the sample standard deviation and n is the number of values in the sample.

If we know the population standard departure, $- \sigma -$, then we tin utilise the normal distribution instead of the t-distribution and the known $- \sigma -$ value instead of the estimate due south.

I have long wondered that if we know the population standard deviation we probably also know the population mean and why would nosotros be calculating a confidence interval. Not sure of a situation where I would use the known standard departure, if you lot know of a situation, do permit me know.

The formula for the confidence interval about a sample mean with a known population standard difference is

$$ \big\displaystyle \bar{Ten}\pm {{Z}_{\blastoff /two}}\frac{\sigma }{\sqrt{due north}}$$

Where $- {{Z}_{\blastoff /2}} -$ is the number of standard deviations from the center to the betoken when the area under the standard normal equals the $- i-{\alpha }/{ii}\; -$ percentile.

Example

Let say we have 4 readings of tensile strength of new polymer alloy based on polypropylene, 32.three, 34.7, 32,vi, and 33.5 MPa. Determine the estimate of the population mean and a 95% confidence interval about the signal estimate.

The mean of the four values is 33.275 Mpa. This is the point estimate of the mean.

The sample standard deviation, s, is one.08 MPa, which nosotros will need for the conviction interval calculation.

$$ \large\displaystyle \bar{X}\pm {{t}_{\alpha /2,\text{ }n-one}}\frac{southward}{\sqrt{north}}=33.275\pm 3.182\left( \frac{1.08}{2} \right)=31.56\text{ and }34.99$$

The critical value is from the t-distribution tabular array with n-i = three degrees of freedom, and $- \alpha -$ = 0.05 / two = 0.025 Practise problems similar this using the same reckoner and references you wait to use during the examination.

Statistical Confidence (commodity)

Reliability with confidence (article)

Tolerance Intervals for Normal Distribution Based Set of Data (commodity)

Source: https://accendoreliability.com/point-and-interval-estimates/#:~:text=A%20point%20estimate%20of%20the,by%20the%20number%20of%20values.

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