For Exercises 5 through 20, assume that the variables are...

For Exercises 5 through $20,$ assume that the variables are normally or approximately normally distributed. Use the traditional method of hypothesis testing unless otherwise specified. Transferring Phone Calls The manager of a large company claims that the standard deviation of the time (in minutes) that it takes a telephone call to be transferred to the correct office in her company is 1.2 minutes or less. A random sample of 15 calls is selected, and the calls are timed. The standard deviation of the sample is 1.8 minutes. At $\alpha=0.01$, test the claim that the standard deviation is less than or equal to 1.2 minutes. Use the $P$ -value method.

For Exercises 5 through $20,$ assume that the variables are normally or approximately normally distributed. Use the traditional method of hypothesis testing unless otherwise specified.
Transferring Phone Calls The manager of a large company claims that the standard deviation of the time (in minutes) that it takes a telephone call to be transferred to the correct office in her company is 1.2 minutes or less. A random sample of 15 calls is selected, and the calls are timed. The standard deviation of the sample is 1.8 minutes. At $\alpha=0.01$, test the claim that the standard deviation is less than or equal to 1.2 minutes. Use the $P$ -value method.

Key Concepts

Test Statistic for Variance

When testing claims about a population variance or standard deviation, the chi-square test statistic is utilized. The test statistic is calculated by multiplying the sample variance by a factor that involves the sample size, and then dividing by the hypothesized variance. This value is then compared to the chi-square distribution with the appropriate degrees of freedom.

Chi-Square Distribution

The chi-square distribution is a probability distribution that is widely used in inferential statistics, particularly for tests involving variance or standard deviation. It is used in tests that compare the variability of a sample to a known standard, assuming that the underlying data are normally distributed.

P-value

The p-value is the probability, under the assumption that the null hypothesis is true, of observing a test statistic as extreme as, or more extreme than, the value observed in the sample. In hypothesis testing, if the p-value is less than or equal to the chosen significance level (?), the null hypothesis is rejected.

Degrees of Freedom

Degrees of freedom refer to the number of values in a calculation that are free to vary once certain constraints are applied. In the context of a variance test using the chi-square distribution, the degrees of freedom are typically equal to the sample size minus one. They determine the shape of the chi-square distribution and are essential for correctly determining critical values and p-values.

Null and Alternative Hypotheses

The null hypothesis (H0) is a statement of no effect or no difference and serves as the default or baseline assumption in statistical testing. The alternative hypothesis (Ha) is what the researcher wants to provide evidence for, and it represents a deviation from the null hypothesis. The outcome of a hypothesis test depends on whether the evidence is strong enough to reject the null hypothesis in favor of the alternative.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to support a claim or hypothesis about a population parameter. It involves formulating a null hypothesis (H0) and an alternative hypothesis (Ha), then using sample data to determine which hypothesis is more consistent with the observed results.

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about the mean, with data near the mean more frequent in occurrence than data far from the mean. It is a key concept in statistics because many statistical tests and confidence interval calculations assume that the data follow a normal distribution or are approximately normal.

Significance Level

The significance level, denoted by ?, is the probability threshold below which the null hypothesis is rejected. It represents the risk of committing a Type I error, which is rejecting the null hypothesis when it is actually true. Typical values for ? are 0.05, 0.01, or 0.10.

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