I apologize for having a very basic question that I've not been able to find the answer to. Using a cumulative probability table, I find that the cumulative probability of flipping 7 or fewer heads in 23 coin tosses ≈ 0.0465. However, I fail to reproduce this figure when I try to calculate the probability manually with a z table.
The standard deviation of a binomial distribution is sqrt(np(1-p)), in this case sqrt(23 x .5 x .5) ≈ 2.3979. The mean of the distribution is np = 11.5, so the z score of flipping 7 heads is (7 - 11.5) / 2.3979 ≈ -1.8766.
Looking up -1.8766 in a z table, I find that the p value of it ≈ 0.0307. My understanding is that the p value reflects the cumulative probability of obtaining a result less than or equal to the one that many (-1.8766) standard deviations away from the mean; thus, why is this p value so different from the 0.0465 that I found in the cumulative probability table?
I know that I'm just misunderstanding something very fundamental, but I've been reading all up on z tables and just can't put my finger on it. Thanks so much for any help!
Perhaps the difference you're seeing is due to the continuity correction. Did you do the calculation with the continuity correction? In any case a p-value is a tail probability, usually one minus a cumulative for right tailed, but could also involve adding up both tails.