Eight coins are tossed at a time,256 times Number of heads observed at each throw is recorded and the results are given below. Find the expected frequencies What are the theoretical values of main and standard deviation? Calculate also the mean the S.D. of the observed frequencies.
No. of heads at throw
0
1
2
3
4
Frequency
2
6
30
52
67
No. of heads at throw
5
6
7
8
Frequency
56
32
10
1
To find the expected frequencies, we need to calculate the theoretical probabilities for each number of heads.
The total number of possible outcomes when 8 coins are tossed is 2^8 = 256.
To calculate the theoretical probabilities, divide the number of ways each number of heads can occur by the total number of possible outcomes:
No. of heads at throw: 0
Number of ways: 2C0 (2 choose 0) = 1
Theoretical probability: 1/256
No. of heads at throw: 1
Number of ways: 8C1 (8 choose 1) = 8
Theoretical probability: 8/256 = 1/32
No. of heads at throw: 2
Number of ways: 8C2 (8 choose 2) = 28
Theoretical probability: 28/256 = 7/64
No. of heads at throw: 3
Number of ways: 8C3 (8 choose 3) = 56
Theoretical probability: 56/256 = 7/32
No. of heads at throw: 4
Number of ways: 8C4 (8 choose 4) = 70
Theoretical probability: 70/256 = 35/128
No. of heads at throw: 5
Number of ways: 8C5 (8 choose 5) = 56
Theoretical probability: 56/256 = 7/32
No. of heads at throw: 6
Number of ways: 8C6 (8 choose 6) = 28
Theoretical probability: 28/256 = 7/64
No. of heads at throw: 7
Number of ways: 8C7 (8 choose 7) = 8
Theoretical probability: 8/256 = 1/32
No. of heads at throw: 8
Number of ways: 8C8 (8 choose 8) = 1
Theoretical probability: 1/256
Now we can calculate the expected frequencies by multiplying the theoretical probabilities by the total number of trials (256):
No. of heads at throw: 0
Expected frequency: (1/256) * 256 = 1
No. of heads at throw: 1
Expected frequency: (1/32) * 256 = 8
No. of heads at throw: 2
Expected frequency: (7/64) * 256 = 28
No. of heads at throw: 3
Expected frequency: (7/32) * 256 = 56
No. of heads at throw: 4
Expected frequency: (35/128) * 256 = 70
No. of heads at throw: 5
Expected frequency: (7/32) * 256 = 56
No. of heads at throw: 6
Expected frequency: (7/64) * 256 = 28
No. of heads at throw: 7
Expected frequency: (1/32) * 256 = 8
No. of heads at throw: 8
Expected frequency: (1/256) * 256 = 1
To calculate the mean of the observed frequencies:
Sum of (No. of heads * Frequency) / Total trials.
Mean = (0*2 + 1*6 + 2*30 + 3*52 + 4*67 + 5*56 + 6*32 + 7*10 + 8*1) / 256
Mean = 862 / 256
Mean = 3.371
To calculate the standard deviation of the observed frequencies:
1. Find the squared differences between each observed frequency and the mean.
2. Sum up the squared differences.
3. Divide the sum by the total number of trials.
4. Take the square root of the result.
Step 1:
Calculate the squared differences between each observed frequency and the mean:
For No. of heads = 0: (0 - 3.371)^2 * 2
For No. of heads = 1: (1 - 3.371)^2 * 6
For No. of heads = 2: (2 - 3.371)^2 * 30
For No. of heads = 3: (3 - 3.371)^2 * 52
For No. of heads = 4: (4 - 3.371)^2 * 67
For No. of heads = 5: (5 - 3.371)^2 * 56
For No. of heads = 6: (6 - 3.371)^2 * 32
For No. of heads = 7: (7 - 3.371)^2 * 10
For No. of heads = 8: (8 - 3.371)^2 * 1
Step 2:
Sum up the squared differences:
Sum = (0.670^2 * 2) + (2.371^2 * 6) + (1.371^2 * 30) + (0.371^2 * 52) + (0.629^2 * 67) + (1.629^2 * 56) + (1.629^2 * 32) + (3.629^2 * 10) + (4.629^2 * 1)
Step 3:
Divide the sum by the total number of trials:
Standard Deviation = sqrt(Sum / 256)
Calculate the square root of the result to get the standard deviation.
To find the expected frequencies, we can start by finding the probability of getting each number of heads.
When eight coins are tossed, the probability of getting 0 heads is the same as getting 8 tails. The probability of getting a tail on a single coin toss is 1/2, so the probability of getting all tails is (1/2)^8 = 1/256.
Similarly, the probability of getting 1 head can be calculated. There are 8 possible ways to get exactly 1 head: H-T-T-T-T-T-T, T-H-T-T-T-T-T, and so on. Each coin toss has a probability of 1/2, so the probability of getting exactly 1 head is 8 * (1/2)^8 = 8/256 = 1/32.
Using the same logic, we can calculate the probabilities for 2, 3, 4, 5, 6, 7, and 8 heads.
No. of heads: Probability: Expected Frequency:
0 1/256 (1/256) * 256 = 1
1 1/32 (1/32) * 256 = 8
2 8/256 (8/256) * 256 = 8
3 28/256 (28/256) * 256 = 28
4 70/256 (70/256) * 256 = 70
5 56/256 (56/256) * 256 = 56
6 32/256 (32/256) * 256 = 32
7 10/256 (10/256) * 256 = 10
8 1/256 (1/256) * 256 = 1
The theoretical values of the mean and standard deviation can be calculated using the expected frequencies. The mean can be calculated as the sum of (frequency * number of heads) divided by the total frequency (256 times):
Mean = (0 * 2 + 1 * 6 + 2 * 30 + 3 * 52 + 4 * 67 + 5 * 56 + 6 * 32 + 7 * 10 + 8 * 1) / 256
To calculate the standard deviation, we need to find the variance first. Variance is calculated as the sum of ((number of heads - mean)^2 * frequency) divided by the total frequency (256 times):
Variance = ((0 - mean)^2 * 2 + (1 - mean)^2 * 6 + (2 - mean)^2 * 30 + (3 - mean)^2 * 52 + (4 - mean)^2 * 67 + (5 - mean)^2 * 56 + (6 - mean)^2 * 32 + (7 - mean)^2 * 10 + (8 - mean)^2 * 1) / 256
The standard deviation is the square root of the variance:
Standard deviation = sqrt(Variance)
Now you can substitute the values from the given table into the formulas to calculate the mean and standard deviation of the observed frequencies.