Question

Statistics with Applications. 1. A random sample of 37 first graders who participated in sports had manual dexterity scores with mean 32.19 and standard deviation 4.34. An independent sample of 37 first graders who did not participate in sports has manual dexterity scores with mean 31.68 and standard deviation 4.56. (a) Test whether there is sufficient evidence to conclude that first graders who participated in sports have a higher dexterity score at the 0.05 level. (b) For the rejection region used in (a), calculate the type II error probability B when u1 - U2 = 3 (i.e., the dexterity score mean of those who participated in sports is 3 points higher).

Answer 1

Answer:

Given that:

Let $Y_1$ denote the manual dexterity scores of second graderas participated

Let $Y_2$   denote the manual dexterity scores of second graderas did not participated in sports had manual dexerity scores

Let $\bar{Y_1}$ be the sample mean of second graders participated in sports

Let $\bar{Y_2}$ be the sample mean of second graders did not participated in sports

Let $s_1$ be the standard deviation of second graders participated in sports

Let $s_2$   be the standard deviation of second graders participated in sports

Let $n_1$ denote number of second graders participated in sports

Let $n_2$ denote number of second graders participated in sports

The given information is as shown below

Second graders who participated in sports

The number of graders $n_1=37$

Sample mean $\bar{Y_1}=32.19$

Sample variance $s_1=4.34$

Second graders who did not participated in sports

The number of graders $n_2=37$

Sample mean $\bar{Y_2}=31.68$

Sample variance $s_2=4.56$

a)

The objective of the study is to test and determine whether there is sufficient evidence to support that second graders who participate in sports have a higher mean dextenity score

Let $\mu _1$ denote the population mean dexterity score for second graders who participated in sports

Let $\mu _2$ denote the population mean dexterity score for second graders who did not participated in sports

The null and alternative hypothesis are

$H_0: \mu _1=\mu _2$

$H_a: \mu _1>\mu _2$

Use the level of significance

= 0,05

From the standard normal distribution tables the critical value at 0.05 level for one tailed is 1.645

Rejection region since the test is based on right tailed reject the null hypothesis if $Z>Z_a(1.645)$

The test statistic is

$Z=\frac{\bar{Y_1}-\bar{Y_2}}{\sqrt{\frac{s_1^2}{n}+\frac{s_2^2}{n}}}$

$Z=\frac{32.19-31.68}{\sqrt{\frac{4.34^2}{37}+\frac{4.56^2}{37}}}$

$Z=\frac{0.51}{1.0349}$

$Z=0.4927$

Here,it can be observed that the test statistic value 0.4927 is less than the critical value 1.645.So the null hypothesis is fails to be rejected

Therefore it can be concluded that there is no sufficient evidence to support that the second graders who participate in sports have a higher dexterity score

b) Find $\beta$ when $\mu _1-\mu -2=3$

From part(a) the rejection region is Z>1.645

Consider $Z\geq 1.645$

$Z=\frac{\bar{Y_1}-\bar{Y_2}}{\sqrt{\frac{s_1^2}{n}+\frac{s_2^2}{n}}}\geq 1.645$

$\bar{Y_1}-\bar{Y_2}\geq 1.645{\sqrt{\frac{4.34^2}{37}+\frac{4.56^2}{37}}}$

$\bar{Y_1}-\bar{Y_2}\geq 1.702$

Implies don't to reject $H_0$ if $\bar{Y_1}-\bar{Y_2}<1.702$

$\beta =P\begin{Bmatrix} Don't\, \, reject\, \, H_o\, \, when\, \, H_1\, \, is \, \, true \end{Bmatrix}$

$=P(\bar{Y_1}-\bar{Y_2}<1.702\mid \mu _1-\mu _2=3)$

$=P(\frac{\bar{Y_1}-\bar{Y_2}-(\mu _1-\mu _2)}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n}}}<\frac{1.702-3}{\sqrt{\frac{4.34^2}{37}+\frac{4.56^2}{37}}})$

$=P(Z\leq -1.25)$

$=0.1056$

Therefore the value of $\beta$ when $\mu _1-\mu _2=3$ is 0.1056