Statistics Examples Normal Distributions

Test the Claim

$n = 49$ , $\overline{x} = 1.71$ , $σ = 0.13$ , $α = 0.05$ , $μ_{\overline{x}} = 0.2$

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.

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

Step 2

Fill in the known values.

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

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

Multiply $- 1$ by $0.2$ .

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

Step 3.3

Subtract $0.2$ from $1.71$ .

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

Step 3.4

Divide $7$ by $0.13$ .

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

Step 3.5

Multiply $1.51$ by $53.\overline{846153}$ .

$81.\overline{307692}$

Step 4

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

$α_{Two Tail} = \frac{α}{2} = 0.025$

Step 5

The critical value represents the z-score that provides a significance level of $α = 0.05$ , since $n > 30$ use the normal distribution.

$z = 2$

Step 6

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

$81.30769 > 2$

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Normal Distributions

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