The number of hours spent watching television by students in...

Name: Problem 55, Chapter 6: Statistics for Business and Economics: Global Edition
Uploaded: 2022-01-12T18:45:54-08:00
Duration: 2 min 4 s
Channel: Nick Johnson
Description: The number of hours spent watching television by students in the week before final exams has a normal distribution with a standard deviation of 4.5 hours. A random sample of 30 students was taken. a. Is the probability more than 0.95 that the sample standard deviation exceeds 3.5 hours? b. Is the probability more than 0.95 that the sample standard deviation is less than 6 hours?

The number of hours spent watching television by students in the week before final exams has a normal distribution with a standard deviation of 4.5 hours. A random sample of 30 students was taken. a. Is the probability more than 0.95 that the sample standard deviation exceeds 3.5 hours? b. Is the probability more than 0.95 that the sample standard deviation is less than 6 hours?

The number of hours spent watching television by students in the week before final exams has a normal distribution with a standard deviation of 4.5 hours. A random sample of 30 students was taken.
a. Is the probability more than 0.95 that the sample standard deviation exceeds 3.5 hours?
b. Is the probability more than 0.95 that the sample standard deviation is less than 6 hours?

Key Concepts

Chi-Square Distribution

The chi-square distribution arises naturally when a number of independent squared standard normal random variables are summed. It is characterized by its degrees of freedom, which in the context of sample variance is typically one less than the sample size (n-1). This distribution is used to assess the probability of observing a sample variance that deviates from the population variance by a certain amount, making it instrumental in tests of variance and in constructing confidence intervals for the variance.

Degrees of Freedom

Degrees of freedom refer to the number of values in a calculation that are free to vary while still satisfying a given constraint. In the context of sample variance for normally distributed data, the degrees of freedom is equal to the sample size minus one (n-1). This parameter is essential in determining the shape of the chi-square distribution, which in turn is used to assess the variability of the sample variance and standard deviation.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its symmetric, bell-shaped curve. It is defined by two parameters, the mean and the standard deviation, and is central in statistics due to the Central Limit Theorem, which states that the sum or average of a large number of independent and identically distributed random variables tends to follow a normal distribution regardless of the original distribution of the variables.

Sampling Distribution of the Variance

When sampling from a normally distributed population, the sample variance is not fixed but has its own probability distribution. Specifically, when scaled appropriately (multiplying by (n-1)/?²), the sample variance follows a chi-square distribution. This distribution helps in understanding the variability of the sample variance around the true population variance, and it is crucial in constructing confidence intervals and conducting hypothesis tests regarding the population variance.

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