Sampling Distribution Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

hard

For a population with

\mu=100

\sigma=20

, and

n=25

: (a) find

P(\bar{X} > 104)

, (b) find the value

c

such that

P(\bar{X} < c) = 0.90

Solution

1
Standard error: $SE = \frac{20}{\sqrt{25}} = 4$
2
(a) $P(\bar{X} > 104) = P\left(Z > \frac{104-100}{4}\right) = P(Z > 1) = 1 - 0.8413 = 0.1587$
3
(b) For $P(\bar{X} < c) = 0.90$ : z-score = 1.282 (from z-table); $c = \mu + z \cdot SE = 100 + 1.282(4) = 105.13$

Answer

(a)

P(\bar{X}>104) \approx 0.1587

; (b)

c \approx 105.13

Sampling distribution calculations convert to z-scores using

z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}

. This is possible because the sampling distribution of

\bar{X}

is approximately normal by CLT. All inference procedures (CIs, tests) are based on this framework.

About Sampling Distribution

The probability distribution of a statistic (such as the sample mean) computed from all possible random samples of the same size drawn from a population.

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