Question

1. Jerry is a district manager for Home Depot and he recently implemented a new safety program in the 8 stores he manages. He would like to know if this new program has resulted in fewer accidents than other Home Depot stores. Jerry knows that the average number of accidents at any Home Depot is 4.2 a month with a standard deviation of 1.1. Jerry’s has the following accident data for his 8 stores. Use a single-sample z-test to determine the outcome (alpha = .05, one-tailed).

Store 1: 2 accidents

Store 2: 1 accidents

Store 3: 4 accidents

Store 4: 3 accidents

Store 5: 5 accidents

Store 6: 1 accidents

Store 7: 0 accidents

Store 8: 2 accidents

            
In the box below, provide the following information:

Null Hypothesis in sentence form (1 point):

Alternative Hypothesis in sentence form (1 point):
Critical Value(s) (2 points):

Calculations (4 points): Note: the more detail you provide, the more partial credit that I can give you if you make a mistake.

Outcome (determination of significance or not, and what this reflects in everyday language, 2 points)

2.

Kroger reports that their full time employees make on average 14.50 dollars per hour. You want to examine if the full time employees at the Richmond store are typical of all Kroger employees in terms of income. The Richmond store has 10 full time employees, and you attained their income below. Use a single-sample t-test to determine the outcome (alpha = .05, two-tailed).

Employee 1:   15.75 per hour

Employee 2:   11.75 per hour

Employee 3:   13.75 per hour

Employee 4:   13.50 per hour

Employee 5:   13.50 per hour

Employee 6:   13.50 per hour

Employee 7:   14.75 per hour

Employee 8:   12.50 per hour

Employee 9:   14.00 per hour

Employee 10: 12.00 per hour

In the box below, provide the following information:

Null Hypothesis in sentence form (1 point):

Alternative Hypothesis in sentence form (1 point):
Critical Value(s) (2 points):

Calculations (4 points): Note: the more detail you provide, the more partial credit that I can give you if you make a mistake.

Outcome (determination of significance or not, and what this reflects in everyday language, 2 points)

Answer 1

1.

Null Hypothesis : Average number of accidents at any Home Depot a month after implementing a new safety program is more than or equal to 4.2 .

Alternative Hypothesis : Average number of accidents at any Home Depot a month after implementing a new safety program is less than 4.2 .

Let average number of accidents at any Home Depot a month after implementing a new safety program be denoted as $\mu$.

Null Hypothesis H₀ : $\mu$=4.2 vs   Alternative Hypothesis H₁: $\mu$<4.2

Let number of accidents at any Home Depot a month after implementing a new safety program be denoted as X.

X ~ N($\mu$,$\sigma$²)

From a sample of size n, let $\bar{X}$ denote the sample mean.

$\bar{X}\sim N(\mu,\sigma^{2}/n)\Rightarrow \frac{\sqrt{n}(\bar{X}-\mu)}{\sigma}\sim N(0,1)$

Given $\alpha$=0.05 ; so for the one-tailed test critical value = -Z_$\alpha$ = -Z_0.05 = -1.645

Calculations :-

Under H₀ ; $\mu$=4.2 , $\sigma$=1.1

$\therefore$ Under H₀ ; $\sqrt{n}(\bar{X}-4.2)/1.1\sim N(0,1)$

Thus the test statistic is Z = $\sqrt{n}(\bar{X}-4.2)/1.1$

Given ; n=8 , $\bar{X}$ = (2+1+4+3+5+1+0+2)/8 = 18/8 = 2.25

$\therefore$ Observed value of Z = $\sqrt{8}(2.25-4.2)/1.1$ = -5.014

Outcome :-

Null Hypothesis is rejected if observed value of the test statistic is less than the critical value.

In this case we see that ; Observed value of Z < Critical value

Thus we reject the hypothesis that average number of accidents at any Home Depot a month after implementing a new safety program is more than or equal to 4.2 .

Thus we can conclude that the recently implemented new safety program has resulted in fewer accidents than other Home Depot stores.

2.

Average income per hour of full time employees at the Kroger store is equal to 14.50 dollars.

Null Hypothesis : Average income per hour of full time employees at the Richmond store equal to 14.50 dollars.

Alternative Hypothesis : Average income per hour of full time employees at the Richmond store not equal to 14.50 dollars.

Let average income per hour of full time employees at the Richmond store be denoted as $\mu$.

Null Hypothesis H₀ : $\mu$=14.50   vs   Alternative Hypothesis H₁: $\mu$$\neq$14.50

Let the income per hour a full time employee at the Richmond store be denoted as X.

X ~ N($\mu$,$\sigma$²)

From a sample of size n, let $\bar{X}$ denote the sample mean an s² denote the sample variance.

$\frac{\sqrt{n}(\bar{X}-\mu)}{s}\sim t_{n-1}$

Given n=10,$\alpha$=0.05 ; so for the two-tailed test critical value = t_{n-1;$\alpha$/2} = t_10-1;0.05/2 = t_9;0.025 = 2.262

Calculations :-

Under H₀ ; $\mu$=14.50

$\therefore$ Under H₀ ; $\sqrt{n}(\bar{X}-14.50)/s\sim t_{n-1}$

Thus the test statistic is t = $\sqrt{n}(\bar{X}-14.50)/s$

$\bar{X}$ = (15.75+11.75+13.75+13.50+13.50+13.50+14.75+15.50+14.00+12.00)/10 = 138/10 = 13.80

s² = (15.75²+11.75²+13.75²+13.50²+13.50²+13.50²+14.75²+15.50²+14.00²+12.00²-10$\times$13.80²)/(10-1) = 1.70556

$\therefore$ Observed value of t = $\sqrt{10}(13.80-14.50)/\sqrt{1.70556}$ = -1.695 $\Rightarrow$|Observed value of Z| = 1.695

Outcome :-

Null Hypothesis is rejected if |observed value of the test statistic| is more than the critical value.

In this case we see that ; |Observed value of Z| < Critical value

Thus we accept the hypothesis that average income per hour of full time employees at the Richmond store is equal to 14.50 dollars.

Thus we can conclude that average income per hour of full time employees at the Richmond store is typical of all Kroger employees.