Question

One sample T hypothesis test:

μ : Mean of variable
H₀ : μ = 50
H_A : μ ≠ 50

Hypothesis test results:

Variable	Sample Mean	Std. Err.	DF	T-Stat	P-value
Optimism Score	44.534	0.33089021	999	-16.519074	<0.0001

1. State the null and alternative hypothesis.

The null hypothesis states that women in poor urban settings are as optimistic as other women nationally.

The alternative hypothesis states that there is a difference in optimism between women in poor urban settings and other women nationally.

2. What is the value of t for the analysis?

T value is -16.519074 or -16.51

3. Is this value significant and at what level? Why?

4. What are the CI values and what do they mean here?

Answer 1

The Value of t is -16.52 (Rounding -16.519)

The p value is < 0.0001. The decision rule is that if the p value is < Alpha, then we Reject H0 (the hypothesis is statistically significant)

In this case the hypothesis is statistically significant at all the usual levels of Alpha (0.10, 0.05, 0.01, 0.001). This is because the p value is very small, which means that the probability of observing a statistic as extreme as or greater than the one obtained is very very low, assuming that the null hypothesis is true.

Since this probability is so small, the chance of the null being true is very small and it gets rejected, resulting in a statistically significant test.

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For the CI

From the data: , = 44.534, SE = 0.33089.

The tcritical (2 tail) for = 0.01, for df = n -1 = 999, is 2.58

The Confidence Interval is given by ME, where

ME = t critical *SE = 2.58 * 0.33089 = 0.854

The Lower Limit = 44.534 - 0.854 = 43.680

The Upper Limit = 44.534 + 0.854 = 45.388

The Confidence Interval is (43.680 , 45.388)

The CI values are 43.680 and 45.388. It means that we are 99% confident that the true population mean lies within the interval limits of 43.680 to 45.388.

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