In 1954, 30% were non-smokers (100-70), while in the recent sample 42% were non-smokers (100-58).
The null hypothesis H0 is that there is no change to the mean μ=30% (non-smokers). Using the 1954 figures the variance, σ2=30×0.70=21, and the standard deviation is √21=4.6% approx. So we estimate that the percentage of non-smokers in 1954=30±4.6 (or the percentage of smokers is 70±4.6). The alternative hypothesis HA is that the new mean for non-smokers is significantly different from the 1954 figure of 30%.
The difference in the means for non-smokers is 42-30=12%. Using the population standard deviation from 1954, we can work out a Z-score=12/4.6=2.61 approx, corresponding to a p-value of 0.9955. The given significance level is 0.01, but for a 2-tail test we split this into 0.005 for each tail which gives us the confidence interval 99%. The p-value is slightly bigger than 0.995 or 99.5%, which means we should reject H0 in favour of HA. We conclude that the mean percentage of non-smokers is different from what it was in 1954. If HA had been that there was a rise in the mean percentage of non-smokers we would have been comparing the p-value with 99% instead of 99.5%, so the results would have been even more significant.