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The 20 data are width to length ratios of beaded rectangles used by the Shoshoni Indians of America to decorate their leather goods.

69.3 65.4 66.2 61.5 69.0 66.8 60.6 60.1 57.0 57.6 74.9 67.0 67.2 60.6 62.8 61.1 60.9 55.3 84.4 93.3

One might ask whether the golden rectangle(for which the width to length ratio is 0.618) can be considered as an aesthetic standard for the ShoShonis just as it was for the Greeks and the Egyptians?

1) I assume null hypothesis Ho: The distribution is same as that of the normal distribution. The normality test gave me p value 0.002, which means I accept Ha(The distribution is not normal)

2) Then i do a one sample t-test with the standard value 0.618 with 95% confidence interval. My Ho:Sample mean and 0.618 are same. The two tailed p value is 0.054.

Now the real question comes what to do now i mean Whether to accept Ho or reject Ho because 0.054 is near to 0.50?

One t-test results

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What the p value tells you is:

If, in the population from which this sample was randomly drawn, the null hypothesis was strictly true, what is the probability of getting a test statistic at least as extreme as the one we got in a sample the size of the one we have.

So, in your case, if this is a random sample of beaded rectangles and if, in the population of all beaded rectangles the actual ratio was exactly the golden ratio, you would only get a t value this high 0.054 of the time.

I don't think this is very useful in your case. For one thing, I think you ought to be doing a test of equivalence, not a test of difference. Given the nature of your data and your question, you could use TOST (two one sided t tests).

Usually, the null hypothesis is the one that we are trying to reject; here, it is the one you are trying to accept.

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  • $\begingroup$ Glad you liked it. The usual thing to do is to upvote answers you like and to accept them if they fully answer your question. (Just putting this here because you are new - welcome to the site!) $\endgroup$
    – Peter Flom
    Commented Sep 8, 2017 at 13:45
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    $\begingroup$ Your remarks about the null seem contrary to standard practice. The scientist is looking for any evidence that might indicate a hypothesis is false. Thus, the correct null ought to be an expression of the idea that this tribe tended to produce, on average, aspect ratios close to the Golden Ratio. You are nevertheless correct to issue words of caution, but for the entirely different reason that the arithmetic mean of a widely varying range of aspect ratios would seem to say almost nothing meaningful about an "aesthetic standard." The distribution of those ratios is of far more interest. $\endgroup$
    – whuber ♦
    Commented Sep 8, 2017 at 15:14

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