Statistics Examples

n = 49

$n = 49$ ,

\overline{x} = 1.71

$\overline{x} = 1.71$ ,

σ = 0.13

$σ = 0.13$ ,

α = 0.05

$α = 0.05$ ,

μ_{\overline{x}} = 0.2

$μ_{\overline{x}} = 0.2$

Step 1

The z-score converts a non-standard distribution to a standard distribution in order to find the probability of an event. For the z-score of the distribution of means, the standard deviation is divided by the square root of the sample size.

\frac{\overline{x} - µ_{\overline{x}}}{\frac{σ}{\sqrt{n}}}

$\frac{\overline{x} - µ_{\overline{x}}}{\frac{σ}{\sqrt{n}}}$

Step 2

Fill in the known values.

\frac{1.71 - 0.2}{\frac{0.13}{\sqrt{49}}}

$\frac{1.71 - 0.2}{\frac{0.13}{\sqrt{49}}}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

(1.71 - (0.2)) \frac{7}{0.13}

$(1.71 - (0.2)) \frac{7}{0.13}$

Step 3.2

Multiply

- 1

$- 1$ by

0.2

$0.2$ .

(1.71 - 0.2) \frac{7}{0.13}

$(1.71 - 0.2) \frac{7}{0.13}$

Step 3.3

Subtract

0.2

$0.2$ from

1.71

$1.71$ .

1.51 (\frac{7}{0.13})

$1.51 (\frac{7}{0.13})$

Step 3.4

Divide

7

$7$ by

0.13

$0.13$ .

1.51 \cdot 53.\overline{846153}

$1.51 \cdot 53.\overline{846153}$

Step 3.5

Multiply

1.51

$1.51$ by

53.\overline{846153}

$53.\overline{846153}$ .

81.\overline{307692}

$81.\overline{307692}$

81.\overline{307692}

$81.\overline{307692}$

Step 4

Since the claim is for an exact value of the mean, use the two tailed test.

α_{Two Tail} = \frac{α}{2} = 0.025

$α_{Two Tail} = \frac{α}{2} = 0.025$

Step 5

The critical value represents the z-score that provides a significance level of

α = 0.05

$α = 0.05$ , since

n > 30

$n > 30$ use the normal distribution.

z = 2

$z = 2$

Step 6

Since the z-score of the test statistic is less than the critical value, there is sufficient evidence to support the hypothesis.

81.30769 > 2

$81.30769 > 2$

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