![\"\"](\"/images/stars/stars-07.gif\")
You can put this solution on YOUR website!
Our null hypothesis is that the coin is not biased towards heads (this is the hypothesis we will attempt to disprove), and our alternate hypothesis is that the coin is biased towards heads. To determine whether or not we may reject our null hypothesis, we first determine the expected values of the experiment: this is the number of heads and number of tails you would expect to get from a fair coin flipped sixty times. Because a fair coin implies it is as likely to come up heads as it is to come up tails, then we expect to have 30 heads and 30 tails when a coin is flipped 60 times. For clarity, we now construct the following table (note that because tails and heads are mutually exclusive, we assume 38 heads implies 60-38= 22 tails):\r
\n" ); document.write( "\n" ); document.write( "
| Observed | Expected | |
| Heads | 38 | 30 |
| Tails | 22 | 30 |
\n" ); document.write( "\n" ); document.write( "We now use the chi-square test to determine whether our 38 heads is simply a coincidence, or implies that the coin is biased towards heads.\r
\n" ); document.write( "\n" ); document.write( "In order to obtain a p value via the chi-square test, we must first determine the chi-square statistic, which follows this eqution: Chi-Square Stat =
\n" ); document.write( "\n" ); document.write( "We must now consult the chi-squre distribution to obtain the corresponding p-value. This is usually done using a table that you will most likely find in the front or back of your text book (usually in an appendix, or on an inside cover). In order to get the p-value, we must use the chi-squre distribution for our number of degrees of freedom. The degrees of freedom is equal to possible outcomes - 1. In our case we have two possible outcomes (heads or tails), so our df = 2 - 1 = 1. So we have 1 degree of freedom. We now look at the table in our textbook in the row for 1 degree of freedom, and find the two cells that lie below and above 4.266667, our chi-square statistic. Your table will probably indicate that the p-value lies between .05 and .025. This is sufficient for our purposes because we need only determine the signficance at a level of .1. We know the p-value must be below .05, so we can reject the null hypothesis that the coin is not biased towards heads (because the p-value is less than the .1, the level of significance).\r
\n" ); document.write( "\n" ); document.write( "It is possible to obtain an exact p-value. This would be accomplished using the chi-square distribution, however this is somewhat complex. More simply, you can derive a p-value through software such as excel using the CHITEST formula. This method indicates a p-score of about .0389. Note that this is NOT NECESSARY because our table has already indicated a p-value between .05 and .025, however it is nice to have an exact value :)\r
\n" ); document.write( "\n" ); document.write( "We interpret the p-value as this: the probability that we rejected the null hypothesis when in fact it is correct. In other words, there is a .037 chance that your conclusion to reject the null hypothesis is incorrect.\r
\n" ); document.write( "\n" ); document.write( "So, in summary, our chi-square test indicates a p-value somewhere between .05 and .025, which is less than the .10 level of significance, so we reject the null hypothesis that the coin is not biased towards heads (loosely, we conclude that the coin is biased towards heads). \n" ); document.write( "