Question

1.3.24 One of the authors read somewhere that it's been conjectured that when people are asked to choose a number from the choices 1,2,3, and 4, they tend to choose "3" more than would be expected by random chance To investigate this, she collected data in her class. Here is the table of responses from her students: Chose 1 Chose 2 Chose 3 Chose 4 10 4 14 Suppose that you wanted to investigate whether people tend to pick a "big" number (3 or 4) rather than a "small" number (1 or 2) Which of the following best describes the null hypothesis and the alternative hypothesis using the appropriate symbol? null: π 0.5, alternative: π < 0.5 O null: p 0.5, alternative: p < 0.5 --, null: π = 0.5, alternative: π > 0.5 nll: 0.5, alternative:p>0.5 eTextbook and Media

Answer 1

Solution:-

a) State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: $\pi$< 0.50
Alternative hypothesis: $\pi$ > 0.50

Note that these hypotheses constitute a one-tailed test.

b)

$\hat{p}=\frac{19}{33} = 0.58$

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method, shown in the next section, is a one-sample z-test.

Analyze sample data. Using sample data, we calculate the standard deviation (S.D) and compute the z-score test statistic (z).

c)

The mean is 0.50.

S.D = sqrt[ P * ( 1 - P ) / n ]

S.D = 0.08704
d)

z = (p - P) / S.D

z = 0.870

where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and n is the sample size.

Since we have a one-tailed test, the P-value is the probability that the z-score is greater than 0.870.

Thus, the P-value = 0.192.

Interpret results. Since the P-value (0.192) is greater than the significance level (0.05), we have to accept the null hypothesis.

We have little to no evidence that the long run proportion of people who will pick a big number(3 or 4) is greater than 50%.