I've calculated the expected frequencies by using the normal distribution chart and the frequencies given in this table. However the sum of the expected frequencies is turning out to be 67 while the sum of the given frequencies in 100. I've redone the calculations only to arrive at the same answer. Is there something wrong with my method or the data is not good for a normal distribution fit? Here is the method. Please help.
(1) An interval like "60-62" is usually interpreted as covering all numbers greater than or equal to 60 but strictly less than 63. Thus the midpoints are 61.5, 64.5, etc.
(2) Your calculation omits the intervals $(-\infty,60),$ $(62,63),$ $(65,66),$ $(68,69),$ $(71,72),$ and $(74,\infty).$ As shown at https://stats.stackexchange.com/a/117711/919 "Method 8," this will cover only about two-thirds of the total probability. Indeed, your calculation of $67/100$ is exactly $2/3$ to the significance given.
Your calculations otherwise look correct, so I won't comment on those, but offer some R code to check. First, here are plots of the results. The black curve is obtained with your moment-mapping estimates while the red curve is adjusted with Sheppard's corrections.
x <- c(60, 63, 66, 69, 72, 75) # Cut points
q <- c(5, 18, 42, 27, 8, 0) # Frequencies
h <- 3 # Bin width
x.mid <- x + h/2 # Midpoints
m <- sum(q * x.mid) / sum(q) # Estimated mean
v <- sum(q * x.mid^2) / sum(q) - m^2
s <- sqrt(v) # Estimated sd
# s <- sqrt(v - h^2/12) # Better: See https://stats.stackexchange.com/a/68238/919
y <- c(x, Inf) # Right endpoints
x <- c(-Inf, x) # Left endpoints
(round(rbind(Start = x, End = y,
Expected frequency = sum(q) * (pnorm(y, m, s) - pnorm(x, m, s))),
1))
Output:
[,1] [,2] [,3] [,4] [,5] [,6] [,7] Start -Inf 60.0 63.0 66.0 69.0 72.0 75.0 End 60.0 63.0 66.0 69.0 72.0 75.0 Inf Expected frequency 0.3 4.2 20.7 38.8 27.7 7.5 0.8
