The 102 students in a class made blinded evaluations of...

The 102 students in a class made blinded evaluations of pairs of cola drinks. For the comparison, cola A was preferred 61 times. In the population that this sample represents, is this strong evidence that a majority prefers one of the drinks? Refer to the following technology output. Complete parts (a) through (d) below. Hypothesis Test: Population Parameter Proportion p Null Hypothesis $H_o: p = 0.50$ Alternative Hypothesis $H_a: p \neq 0.50$ a. The sample proportion falls how many standard errors from the value of p under the null hypothesis? (Round to two decimal places as needed.) Test Statistic z 1.98 P- Value 0.0477

Transcript

00:01 Okay, so we've got a sample of size 100 and 57 of them say they prefer brand a.

00:09 So that's a proportion of 0 .57 in a sample that prefer brand a.

00:13 So it wants a hypothesis to conduct a hypothesis test where the null hypothesis is that the population proportion actually equals some value p -0, which they want us to check the values from 0 .46.

00:31 All the way up to 0 .68.

00:34 And the alternative hypothesis is that this isn't true.

00:38 So p is not equal to p .0.

00:41 Alpha equals 0 .05.

00:43 Now, because this is a not equal to, we've got a two -sided hypothesis test.

00:48 And so the critical z value at the 0 .05 level is going to be the z value associated with 97 .5, because we're after the central 95%, and so this said value corresponds to 97 .5%.

01:08 And that is 1 .96.

01:12 So that's our critical value.

01:14 And then we need to compute test statistics.

01:16 So the test statistic here for a proportion is given by the sample proportion minus the proportion as in the null hypothesis, this divided by the standard deviation.

01:33 Now we don't know the population standard deviation, so we're going to have to use a sample standard deviation.

01:38 And for a proportion where we know the sample proportion, sorry, for a sample where we know the sample proportion, it's going to be p hat times 1 minus p hat divided by the sample size like so.

01:54 And so if you compute these z values, for various p values, you can find that, for example, for 0 .67, it's minus 2 .13, which is less than minus z critical, and so it's significant.

02:20 But for 0 .66, we find it's minus 1 .9, which is greater than minus z critical, so it's not in the critical region, so it's not significant.

02:32 Similarly, at the other end, you find that 0 .48 gives you 1 .8, which is less than z critical, so it's not in a critical region.

02:44 So remember, minus z critical is here, plus z critical is here.

02:50 Terrible.

02:50 But you know.

02:53 And so minus 2 .13 is in the critical region because it's there, minus 1 .9 is there, so it's not in the critical region.

03:02 So 0 .48 is fine, but 0 .47 will give us 2 .0, which is in the critical region.

03:13 It's over here.

03:15 So we do not reject h0.

03:23 For 0 .48 all the way up to 0 .66.

03:30 And the values of p0 represent the proposed population proportion...

Question

Please give Ace some feedback