A coin was tossed 400 times and head appeared 221 times. Test the hypothesis that the coin is unbiased.

Answer 1

What does it mean for a two-sided coin to be unbiased? It is expected of a fair coin to show heads and tails with equal frequency (50%). This hypothesis test is for a single proportion, the proportion of heads appearing on a coin toss out of several tosses. Formally, the test reads like this:

H₀: p = 0.5

H₁: p ≠ 0.5

Think about it: If the coin is biased, it will show fewer of one side (and therefore more of the other). We haven't specified if it will be more heads or more tails that appear, for example. We just care that the sides of the coin are not showing equally, or 50% of the time for each.

We want to show that the number of times that a certain side shows (out of all flips) is equal to the number of appearances from the other side (out of all flips). So, what we want to show is actually in the null hypothesis. (It usually shows in the alternative hypothesis, but in this case, we are showing equality, which is shown in the null hypothesis.)

Decide on a level of significance. Unfortunately, the original question does not say what level of significance to use. The two most common values, α = 0.01 and α = 0.05, will give different outcomes. I am picking α = 0.05, and this is purely my choice. If we get a p-value less than significance, we will reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

Now, we need our sample proportion, where x is the number of heads: x / n = 221 / 400 = 0.5525.

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If you are doing this problem on a TI-83 or TI-84 calculator, go to STAT --> TESTS --> 5: 1-PropZTest.

You need the hypotheses above in order to tell the calculator what to do. Here, p₀ = 0.5, x = 221, and n = 400. Also, as the alternative hypothesis has ≠ , use this option on the line where it says "prop." Then hit "Calculate." You have a z test statistic of 2.1 and a p-value of 0.0358. (The next section is for manual methods; jump down to the Results section of this response.)

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If you are doing this problem manually, you will need this formula for a z-statistic:

z = (p-hat - p₀) / sqrt(p₀q₀/n), where q₀ = 1 - p₀.

z = (0.5525 - 0.5) / sqrt((.5*.5)/400)

z = 0.0525 / 0.025

z = 2.1

You would then need to find the area to the right of this value by finding the area below it (find z = 2.1 in a z-table) and then subtract that value from 1. You should have 1 - 0.9821 = 0.0179. However, since this is a two-tailed hypothesis test (our alternative hypothesis features ≠ ), you need to multiply this number by 2. This becomes 2(0.0179) = 0.0358. This is the p-value.

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Results

Going off of the α = 0.05 level of significance, our p-value is less than this, so we reject the null hypothesis since we have sufficient evidence to conclude that the proportion of heads is different from the proportion of tails for this coin.

Now let's say that we had the α = 0.01 level of significance. Our p-value (still the same) would actually be greater than the significance level, and we would fail to reject the null hypothesis. However, this doesn't mean that we accept the hypothesis that the coin shows heads and tails in equal proportions; it simply means that we don't have sufficient evidence to conclude that this proportion is different from 0.5.