The following relative frequency distribution was constructed from a population of 400. Calculate the population mean, the population variance, and the population standard deviation. (Negative values should be indicated by a minus sign. Round intermediate calculations to at least 4 decimal places and final answers to 2 decimal places.) Class Relative Frequency −20 up to −10 0.22 0.22 −10 up to 0 0.340.32 0 up to 10 0.380.34 10 up to 20 0.06

For the variance first we need to find the second moment given by:

And if we replace we got:

And then the variance is given by:

And the deviation is given by:

Step-by-step explanation:

Previous concepts

In statistics and probability analysis, the expected value "is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur and then summing all of those values".

The variance of a random variable Var(X) is the expected value of the squared deviation from the mean of X, E(X).

And the standard deviation of a random variable X is just the square root of the variance.  

Solution to the problem

Let X our random variable of interest.

For this case we can calculate the mean , variance and deviation creating the following table:

Class          Midpoint(xi)    fi           xi*fi     xi^2 *fi

-20 to-10         -15           0.22        -3.3      49.5

-10-0                 -5           0.34        -1.7        8.5

0-10                   5           0.38         1.9        9.5

10-20                15           0.06        0.9       13.5

And for this case we can find the expected value with the following formula:

And if we replace we got:

For the variance first we need to find the second moment given by:

And if we replace we got:

And then the variance is given by:

And the deviation is given by:

The null and alternative hypotheses:

H0: attitudes between first survey and second survey have not changed.

H1:attitudes between first survey and second survey have changed.

Let's find the observed and expected value using the following:

Observed value = sample size * observed proportion

Expected value = sample size * expected proportion.

Optimistic:

expected= 6%*380=22.8

observed=1%*380=3.8  

(fo-fe)²/fe = 15.83333

Slightly Optimistic:

expected: 29%*380=110.2);

Observed: 30%*380= 114

(fo-fe)²/fe  = 0.131034

Slightly Pessimistic:

expected= 24%*380= 91.2

Observed = 28%*380=106.4  

(fo-fe)²/fe =  2.533333

Pessimistic:

Expected = 41%*380 =155.8

Observed = 41%*380 =155.8

(fo-fe)²/fe = 0

Total (fo-fe)²/fe =

15.83333 + 0.131034 + 2.533333 + 0 = 18.4977

Degrees of freedom, df = 4 - 1 = 3

For chi square test statistic:

Σ(fo-fe)²/fe = 15.83333 + 0.131034 + 2.533333 + 0 =  18.498

Calculate pvalue.

Calculating for pvalue at, df = 3, chi-square score = 18.4977, α=0.05

Pvalue = 0.000347

Decision:

Since pvalue, 0.000347 is less than significance level, 0.05, reject null hypothesis H0.

Conclusion:

There is enough evidence to conclude that there is a change in attitudes between  first survey and the second survey at a level of significance, 0.05.