Answer to Question #211063 in Statistics and Probability for Millicent

Question

N&M magazine believes that it has a 38% share of the national female readership market of women’s magazines. When 2 000 readers of women’s magazines were randomly selected and interviewed, 700 stated that they read N&M regularly.

 (3)

2.1.1Does the sample evidence support their claim? Explain.(5)

2.1.2 What is the population of interest in this case?(2)

2.1.3What is the sample in this case?

(2)

2.1.4 What percentage of readers interviewed read N&M magazine regularly? Is this a statistic or a parameter? Explain.

2.1.5 What does the value 38% represent in the above case? Explain.

Answer 1

2.1.1

$$\hat{p}=\dfrac{700}{2000}=0.35$$

The following null and alternative hypotheses for the population proportion needs to be tested:

$H_0:p=0.38$

$H_1:p\not=0.38$

This corresponds to a two-tailed test, for which a z-test for one population proportion will be used.

Based on the information provided, the significance level is $\alpha=0.05,$ and the critical value for a two-tailed test is $z_c=1.96.$

The rejection region for this two-tailed test is $R=\{z:|z|>1.96\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\hat{p}-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}=\dfrac{0.35-0.38}{\sqrt{\dfrac{0.38(1-0.38))}{2000}}}\approx-2.764$$

Since it is observed that $|z|=2.764>1.96=z_c,$ it is then concluded that the null hypothesis is rejected. Therefore, there is enough evidence to claim that the population proportion $p$ is different than 0.38, at the $\alpha=0.05$ significance level.

Using the P-value approach: The p-value is $p=2P(z<-2.764)=0.005710,$ and since $p=0.005710<0.05=\alpha,$ it is concluded that the null hypothesis is rejected. Therefore, there is enough evidence to claim that the population proportion $p$ is different than 0.38, at the $\alpha=0.05$ significance level.

Therefore, there is enough evidence to claim that the N&M magazine has not a 38% share of the national female readership market of women’s magazines. 

2.1.2. All female magazine readers.

2.1.3. The 2000 randomly selected female magazine readers.

2.1.4. $\dfrac{700}{2000}\cdot100\%=35\%.$ This a sample statistic.

The sample proportion is a random variable: it varies from sample to sample in a way that cannot be predicted with certainty.

2.1.5. 38% represents the hypothesized value in this case.

A hypothesis test formally tests if a population parameter is different to a hypothesized value.