SOLUTION: a claim is made that the proportion of children who play sports is less than .5, and the sample statistics include n=1320 subject with 30% saying that they play a sport

Question 876682: a claim is made that the proportion of children who play sports is less than .5, and the sample statistics include n=1320 subject with 30% saying that they play a sport
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the mean proportion of the population is .5
the null hypothesis is that the population mean proportion is .5.
the alternate hypothesis is that the population mean proportion is less than .5
the mean proportion of the sample is .3
the sample size is 1320
the standard error of the distribution of sample mean proportions is:
sqrt(.3*.7/1320) which is equal to .012613 truncated to 5 decimal places.
.3 is the sample mean proportion.
.7 is 1 minus the sample mean proportion.
the formula is:
se = sqrt(p*(1-p)/n)
se stands for standard error otherwise known as the standard deviation of the distribution of sample means.
a decent reference on all of this can be found here:
http://onlinestatbook.com/2/sampling_distributions/samp_dist_p.html
the z score is (.3 - .5) / .012613 = -15.85665 truncated to 5 decimal places.
the critical alpha at a confidence level of 95% is equal to:
(100% - 95%) / 200 = .025
the critical z score is equal to z(.025) = -1.96
the actual z score of -15.85665 is way beyond that which means the results of the study indicate that the probability that the difference is due to sampling error is so small as to be unmeasurable which means that is virtually no chance that the sample mean of .3 was due to sampling error.
the null hypothesis that the population mean proportion is .5 is rejected in favor of the alternate hypothesis that the population mean proportion is less than .5.

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