Hugger Polls contends that an agent conducts a mean of 53...

Hugger Polls contends that an agent conducts a mean of 53 in-depth home surveys every week. A streamlined survey form has been introduced, and Hugger wants to evaluate its effectiveness. The number of in-depth surveys conducted during a week by a random sample of 15 agents are: $\begin{array}{lllllllllll}53 & 57 & 50 & 55 & 58 & 54 & 60 & 52 & 59 & 62 & 60 & 60 & 51 & 59 & 56\end{array}$ At the .05 level of significance, can we conclude that the mean number of interviews conducted by the agents is more than 53 per week? Estimate the $p$ -value.

Hugger Polls contends that an agent conducts a mean of 53 in-depth home surveys every week. A streamlined survey form has been introduced, and Hugger wants to evaluate its effectiveness. The number of in-depth surveys conducted during a week by a random sample of 15 agents are:
$\begin{array}{lllllllllll}53 & 57 & 50 & 55 & 58 & 54 & 60 & 52 & 59 & 62 & 60 & 60 & 51 & 59 & 56\end{array}$
At the .05 level of significance, can we conclude that the mean number of interviews conducted by the agents is more than 53 per week? Estimate the $p$ -value.

Key Concepts

Hypothesis Testing

Hypothesis testing is a method used in statistics to determine whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. It involves setting up a null hypothesis (usually representing a status quo or a baseline value) and an alternative hypothesis (representing the effect or difference being tested), then using sample data to make a decision regarding these hypotheses.

One-Sample t-Test

The one-sample t-test is used to assess whether the mean of a single sample is statistically different from a known or hypothesized population mean when the population standard deviation is not known. This test is particularly useful for small sample sizes, and it utilizes the t-distribution because it accounts for the extra uncertainty from estimating the population standard deviation from the sample.

Test Statistic

The test statistic in a one-sample t-test is calculated as the difference between the sample mean and the hypothesized population mean, divided by the estimated standard error of the sample mean. This statistic quantifies how far the sample mean is from the null value in units of standard error, and it is used to determine the probability of observing such a result under the null hypothesis.

Degrees of Freedom

Degrees of freedom in the context of a t-test typically refer to the number of values in the final calculation of a statistic that are free to vary. For a one-sample t-test, it is calculated as the sample size minus one. This parameter is crucial for determining the shape of the t-distribution used in calculating the p-value and critical values for the test.

P-Value

The p-value is the probability of obtaining a test statistic as extreme as the one observed, or one more extreme, under the assumption that the null hypothesis is true. It is a measure of the strength of evidence against the null hypothesis: a lower p-value indicates stronger evidence for rejecting the null hypothesis in favor of the alternative hypothesis.

Significance Level

The significance level is a threshold set by the researcher (often denoted by alpha) that defines the maximum acceptable probability of committing a Type I error (rejecting a true null hypothesis). If the p-value is less than or equal to the significance level, the result is considered statistically significant, leading to the rejection of the null hypothesis.

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