Question

Is it possible to not reject both the null hypothesis and alternative hypothesis? It kinda sounds impossible by common sense, so I wonder if there's an error in my working, thanks for your help/
In a condition of n=84, (sample mean) = 0.4302, s (sample STDV) = 0.260, d.f.= 83, level of significance 0.05,
confidence level 95%, critical value (one tailed) 1.663
I have made a null hypothesis and alternative of H₀: μ (the "estimated" population mean) ≤ 0.453 and H₁: μ > 0.453
and by finding the computed value of t, (− μ) ÷ / =-0.8, I concluded not to reject H₀

But what baffled me was when I repeated the same procedure, but now the alternative hypothesis as the new null hypothesis,
thus, H₀: μ ≥ 0.453, H₁: μ< 0, I also could not reject the hypothesis. I know can mathematically understand this because we're using just the same numbers with a different direction of sign of inequality, but I can't understand how both hypotheses can be accepted.

I'll be waiting for any expert's kind replies, thank you for reading.

Answer 1

The choice of Null Hypothesis and Alternative Hypothesis purely depends on the research question on hand. They cannot be just interchanged at will as described in Problem 1 and Problem 2 below:

Problem 1:

In this particular investigation:

The research question is as follows:

Is the population mean greater than 0.453.

For testing the research question:

H0: Null Hypothesis: 0.453 (the population mean is not   greater than 0.453.)

HA: Alternative Hypothesis: 0.453 (the population mean is    greater than 0.453.) (Claim)

n = 84

= 0.4302

s = 0.260

=0.05

df = 84 -1 = 83

From Table, critical value of t = 1.663

Test Statistic is given by:

Since calculated value of t = - 0.804 is less than criticalvalue of t = 1.663, the difference is not significant. Fail to reject null hypothesis.

Conclusion:
The data do not suport the claim that the population mean is    greater than 0.453.

Problem 2:

In this particular investigation:

The research question is as follows:

Is the population mean less than 0.453.

For testing the research question:

H0: Null Hypothesis: 0.453 (the population mean is not less than   0.453.)

HA: Alternative Hypothesis: 0.453 (the population mean is less than   0.453.) (Claim)

n = 84

= 0.4302

s = 0.260

=0.05

df = 84 -1 = 83

From Table, critical value of t = - 1.663

Test Statistic is given by:

Since calculated value of t = - 0.804 is greater than criticalvalue of t = - 1.663, the difference is not significant. Fail to reject null hypothesis.

Conclusion:
The data do not suport the claim that the population mean is less than 0.453.

Thus, we note the following:

The choice of Null Hypothesis and Alternative Hypothesis purely depends on the research question on hand. They cannot be just interchanged at will.

In this particular investigation:

If the research question is:

Is the population mean greater than 0.453.

The conclusion is:

The data do not suport the claim that the population mean is    greater than 0.453.

On the conrary:

If the research question is:

Is the population mean less than 0.453.

The conclusion is:

The data do not suport the claim that the population mean is less than 0.453.