In a certain school district, it was observed that 25% of the students

Question

 in the element schools were classified as only children (no siblings). However, in the special program for talented and gifted children, 88 out of 287 students are only children. The school district administrators want to know if the proportion of only children in the special program is significantly different from the proportion for the school district. Test at the a=0.02 level of significance.

Ho:p=0.25

Ha:p=/0.25

Using the normal approximation for the binomial distribution (without the continuity correction), what is the test statistic for this sample based on the sample proportion?

z= Incorrect

(Report answer as a decimal accurate to 3 decimal places.)

You are now ready to calculate the P-value for this sample.

P-value = Incorrect

(Report answer as a decimal accurate to 4 decimal places.)

These are the correct values how to calculate? since it says keeps saying my answer is wrong?

z= 2.2151940739207

p=0.0267

Answer 1

For a hypothesis test for means or proportions (this one being proportions), to calculate the test statistic (in this case z), we always use the following equation:

z = (sample statistic - population parameter)/standard deviation of the sampling distribution

In this particular problem, the sample statistic is the sample proportion (88/287) and the population parameter is the proportion from the null hypothesis (0.25).

To calculate the standard deviation of the sampling distribution we use the formula sqrt(pq/n). Since this is a hypothesis test, you should be plugging in 0.25 for p here, NOT 88/287. Since p=0.25, q = .75 (because q is always 1 - p) and n is the sample size (be careful here the sample size is 287 NOT 88). This should give you approximately 0.02556.

If we plug that back into the test statistic formula, we get

z = ((88/287) - .25)/0.02556 = 2.215

From there to find the p-value, you would either use a normal distribution table or a calculator (I'm not sure which you are using - if you need more help with this aspect of the problem let me know!) to calculate the area in the tail above the value of 2.215. (This area is approximately 0.013) The key here is that since test is two-tailed (because the alternative hypothesis has =/ rather than < or >) you need to make sure you double the value you get in the tail. So doubling the 0.013 gives a p-value of 0.026. If the alternative hypothesis had simply been < or >, we would not have doubled the 0.013 - that would have been the p-value.

The final step is to use the p-value of 0.026 to draw a conclusion. Since you are using a 0.02 level of significance, we compare the p-value to that value. If the p-value is lower than the level of significance we reject the null hypothesis and if the p-value is higher than the level of significance we fail to reject the hypothesis.

So, in this case, we would fail to reject the null hypothesis because our p-value of 0.026 is greater than the significance level of 0.02.

*There was a lot to unpack with this question... if there is a specific aspect you would like to clarify in more detail, please let me know!*