In a study, researchers wanted to measure the effect of alcohol on the hippocampal region, the portion of the brain responsible for long-term memory storage, in adolescents. The researchers randomly selected 11 adolescents with alcohol use disorders to determine whether the hippocampal volumes in the alcoholic adolescents were less than the normal volume of 9.02 cm3. An analysis of the sample data revealed that the hippocampal volume is approximately normal with no outliers and x=8.06 cm3 and s=0.7 cm3. Conduct the appropriate test at the α=0.01 level of significance. State the null and alternative hypotheses. H0: μ ▼ less than< greater than> equals= not equals≠ nothing H1: μ ▼ less than< not equals≠ equals= greater than> nothing (Type integers or decimals. Do not round.) Identify the t-statistic. t0=nothing (Round to two decimal places as needed.) Identify the P-value. P-value=nothing (Round to three decimal places as needed.) Make a conclusion regarding the hypothesis. ▼ Fail to reject Reject the null hypothesis. There ▼ is is not sufficient evidence to claim that the mean hippocampal volume is ▼ greater than equal to less than nothing cm3

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.3: Least Squares Approximation

Problem 31EQ

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In a study, researchers wanted to measure the effect of alcohol on the hippocampal region, the portion of the brain responsible for long-term memory storage, in adolescents. The researchers randomly selected

adolescents with alcohol use disorders to determine whether the hippocampal volumes in the alcoholic adolescents were less than the normal volume of

9.02

cm3.

An analysis of the sample data revealed that the hippocampal volume is approximately normal with no outliers and

x=8.06

cm3

and

s=0.7

cm3.

Conduct the appropriate test at the

α=0.01

level of significance.

State the null and alternative hypotheses.

H0:

▼

less than<

greater than>

equals=

not equals≠

nothing

H1:

▼

less than<

not equals≠

equals=

greater than>

nothing

(Type integers or decimals. Do not round.)

Identify the t-statistic.

t0=nothing

(Round to two decimal places as needed.)

Identify the P-value.

P-value=nothing

(Round to three decimal places as needed.)

Make a conclusion regarding the hypothesis.

▼

Fail to reject

Reject

the null hypothesis. There

▼

is not

sufficient evidence to claim that the mean hippocampal volume is

▼

greater than

equal to

less than

nothing

cm3

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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