Four coins are tossed 160 times. X denotes the number of heads. The observed&nbsp;frequencies are also given in the following table. Fit a binomial distribution&nbsp;assuming that the coins are unbiased.X: Number of heads 0 1 2 3 4Observed frequency 8 34 69 43 6&nbsp;

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Four coins are tossed 160 times. X denotes the number of heads. The observed&amp;nbsp;frequencies are also given in the following table. Fit a binomial distribution&amp;nbsp;assuming that the coins are unbiased.X: Number of heads 0 1 2 3 4Observed frequency 8 34 69 43 6&amp;nbsp;

karton · Accepted Answer

To fit a binomial distribution, we need to determine the probability of getting X heads out of 4 coin tosses. Assuming that the coins are unbiased, the probability of getting a head on a single coin toss is 0.5, and the probability of getting a tail is also 0.5.Let X be the number of heads in 4 coin tosses. Then, X follows a binomial distribution with parameters n=4 and p=0.5. The probability mass function of X is given by:P(X=k)=(4k)pk(1−p)4−kwhere k=0, 1, 2, 3, 4.To fit the binomial distribution to the observed data, we need to calculate the expected frequencies for each value of X using the above formula. The expected frequency of getting k heads is given by:Ek=n·P(X=k)=4·(4k)0.5k0.54−k=(4k)0.54where k=0, 1, 2, 3, 4.The expected frequencies are:X01234Ek14641Now, we can use the chi-squared goodness-of-fit test to test the goodness of fit of the binomial distribution to the observed data. The chi-squared statistic is given by:χ2=∑k=04(Ok−Ek)2Ekwhere O_k is the observed frequency of getting k heads and E_k is the expected frequency of getting k heads.Substituting the observed and expected frequencies into the formula, we get:χ2=(8−1)21+(34−4)24+(69−6)26+(43−4)24+(6−1)21=115.75;The degrees of freedom for the chi-squared test is given by df = k - 1 = 4 - 1 = 3 (where k is the number of categories).Using a chi-squared distribution table with 3 degrees of freedom and a significance level of 0.05, the critical value is 7.815.Since the calculated chi-squared value (115.75) is greater than the critical value (7.815), we reject the null hypothesis that the observed data follows a binomial distribution with parameters n=4 and p=0.5, and conclude that the coins are biased.

Four coins are tossed 160 times. X denotes

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