According to the records of a past survey conducted among...

According to the records of a past survey conducted among the students of an engineering college, $30 \%$ of the students use their own vehicle to travel to and from college. Recently, 26 students were surveyed at random and it was found that 11 were using their own vehicle for travel. Is it correct to believe that the proportion of students using their own vehicle has increased? Use a 0.05 level of significance.

Key Concepts

Sampling Distribution of Proportions

The sampling distribution of a proportion describes how the sample proportion varies from sample to sample. When the sample size is relatively large (or under conditions where a binomial distribution can be approximated by a normal distribution), this concept is critical in assessing the variability and constructing the test statistic for hypothesis testing.

P-value

The p-value is the probability, given the null hypothesis is true, of obtaining a test statistic at least as extreme as the one observed. A p-value lower than the significance level indicates sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.

Significance Level

The significance level is a threshold set before testing (commonly denoted as ?, like 0.05) that defines how strong the evidence must be to reject the null hypothesis. It represents the probability of committing a Type I error, which is rejecting the null hypothesis when it is actually true.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample to infer that a particular condition holds true for the entire population. It involves formulating a null hypothesis, which represents the status quo, and an alternative hypothesis, which represents the claim to be tested.

One-Tailed Test

A one-tailed test is used when the alternative hypothesis is directional, meaning it tests for an effect in one specific direction only. In this context, it is used to determine whether the proportion of students using their own vehicle has increased from the historical value.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H?) is a statement of no effect or no difference—in this case, that the proportion of students using their own vehicle remains at the historical level. The alternative hypothesis (H?) is a statement that contradicts the null, such as the claim that the proportion has increased.

Test Statistic

A test statistic is calculated from sample data and is used to decide whether to reject the null hypothesis. For proportion tests, this often involves computing a Z statistic based on the difference between the observed sample proportion and the hypothesized population proportion, normalized by the standard error.

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