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Smartphones Many people have trouble setting up all the features of their smartphones, so a company has developed what it hopes will be easier instructions. The goal is to have at least \(96 \%\) of customers succeed. The company tests the new system on 200 people, of whom 188 were successful. Is this strong evidence that the new system fails to meet the company's goal? A student's test of this hypothesis is shown. How many mistakes can you find? $$ \begin{array}{l} \mathrm{H}_{0}: \hat{p}=0.96 \\ \mathrm{H}_{\mathrm{A}}: \hat{p} \neq 0.96 \\ \mathrm{SRS}, 0.96(200)>10 \\ \frac{188}{200}=0.94 ; S D(\hat{p})=\sqrt{\frac{(0.94)(0.06)}{200}}=0.017 \end{array} $$ \(z=\frac{0.96-0.94}{0.017}=1.18\) $$ \mathrm{P}=P(z>1.18)=0.12 $$

Step by step solution

01

Identify the Hypotheses

The hypotheses are correctly identified as the null hypothesis as \(H_0: \hat{p}=0.96\) and the alternative hypothesis as \(H_A: \hat{p} \neq 0.96\) where \(\hat{p}\) denotes the sample proportion.

02

Check Conditions for Testing

The conditions for testing are stated as SRS and \(0.96*200 > 10\), which is correct. SRS ensures the sample data is collected randomly and the second condition checks whether the sample size is large enough for the Central Limit Theorem to apply.

03

Calculate the Sample Proportion

The sample proportion is calculated as \(\hat{p} = \frac{188}{200} = 0.94\), which is correct.

04

Calculate the Standard Deviation of the Sample Proportion

The standard deviation of the sample proportion \(\hat{p}\) is calculated as \(SD(\hat{p})=\sqrt{\frac{(0.94)(0.06)}{200}} = 0.017\). This is incorrect. The calculation of the standard deviation should be based on the null hypothesis proportion and not the sample proportion. So it should be calculated as \(\sqrt{\frac{(0.96)(0.04)}{200}}\)

05

Calculate the Test Statistic

The computation of the test statistic is \(z = \frac{0.96 - 0.94}{0.017} = 1.18\). This is incorrect due to the previously incorrect calculation of the standard deviation.

06

Calculate the P-Value

The P-value is calculated as \(P = P(z>1.18) = 0.12\). This is incorrect as per the incorrect computation of the z-score and thus the resultant P-value.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

The null hypothesis, denoted as \textbf{H₀}, is a statistical assertion that there is no effect or no difference, and it provides a baseline measure that is tested against. In the context of the smartphone company's test, the null hypothesis states that the proportion of customers who succeed with the new instructions is 96%, or \textbf{H₀}: \( \hat{p} = 0.96 \). The null hypothesis is typically the hypothesis that researchers try to disprove, reject, or nullify.

It's crucial to establish a null hypothesis because it allows one to calculate the probability of observing the data if the null hypothesis were true; this calculation involves the P-value, which we will discuss further in another section.

Alternative Hypothesis

Contrasting the null hypothesis, the alternative hypothesis, \textbf{H_A}, suggests that there is a statistically significant effect or difference. For the smartphone company's scenario, the alternative hypothesis posits that the successful setup proportion is not 96%, which is written as \textbf{H_A}: \( \hat{p} eq 0.96 \).

This proposition is what we might believe to be the case if we reject the null hypothesis. It serves as a statement that will be accepted if the null hypothesis is shown to be unlikely.

Sample Proportion

The sample proportion, represented as \( \hat{p} \) , is the ratio of individuals in the sample with the specific characteristic divided by the total number of individuals in the sample. It is a statistical measure used in hypothesis testing. In the smartphone company example, 188 out of 200 participants successfully followed the instructions, thus the sample proportion is calculated as \( \hat{p} = \frac{188}{200} = 0.94 \).

This value is a point estimate of the proportion of the population successfully using the instructions, and plays a crucial role in determining the test statistic.

Standard Deviation of the Sample Proportion

The standard deviation of the sample proportion quantifies how spread out the proportions are in repeated samples from the same population. It's a measure of variability. It is commonly calculated under the assumption of the null hypothesis. Therefore, based on the null hypothesis value of \( \hat{p} = 0.96 \), the standard deviation should be computed as \( SD(\hat{p}) = \sqrt{\frac{(0.96)(0.04)}{200}} \).

The error in the company example was using the sample proportion instead of the null hypothesis proportion. This leads to an incorrect estimation of the variability and thus, the subsequent test statistic and P-value.

Test Statistic

The test statistic is a standardized value that is used to determine how far the sample statistic lies from the null hypothesis. It can be thought of as the number of standard deviations away from what is assumed in the null hypothesis. In the smartphone company case, the test statistic is supposed to be calculated using the corrected standard deviation based on the null hypothesis. Therefore, with the correct standard deviation, the test statistic should be \( z = \frac{0.96 - 0.94}{\text{corrected SD}} \).

The test statistic provides a basis for determining the P-value and consequently, for deciding whether the null hypothesis can be rejected.

P-value

A P-value, or probability value, is a measure used in hypothesis testing to determine the significance of the results. It is the probability of observing a test statistic as extreme as, or more extreme than, the value observed, given that the null hypothesis is true. If this P-value is smaller than a predefined significance level (often 0.05), the results are labelled statistically significant, and the null hypothesis may be rejected. However, in our smartphone example, due to incorrect calculations of the test statistic, the derived P-value was also incorrect. The P-value provides key evidence about the validity of the null hypothesis and requires careful interpretation.

Central Limit Theorem

The Central Limit Theorem (CLT) states that, given a sufficiently large sample size, the sampling distribution of the sample mean will be normally distributed regardless of the distribution of the population from which the sample is drawn. The theorem allows for the simplification of many statistical procedures, including hypothesis testing. The condition specified in the problem, \( 0.96 \times 200 > 10 \), ensures that the sample size is large enough for the CLT to apply, which justifies the use of the normal model for the sampling distribution of \( \hat{p} \).

The theorem emphasizes that the results of the statistical tests relying on normality are valid when the sample size is large, as it is in the smartphone company's case.