Question

A company packages apples by weight. One of the weight grades is Class $A$apples. Class $A$apples have a mean weight of $150g$, and there is a maximum allowed weight tolerance of $5\%$above or below the mean for apples in the same consumer package. A batch of apples is selected to be included in a Class$A$apple package. Given the following apple weights of the batch, does the fruit comply with the Class $A$grade weight tolerance requirements. Conduct an appropriate hypothesis test.

(a) at the $5\%$significance level

(b) at the $1\%$significance level Weights in selected apple batch (in grams): $158$; $167$; $149$; $169$;$164$; $139$; $154$; $150$; 157 ; 171; 152; 161 ; 141; 166; 172;

Short answer

There is insufficient evidence to insure that the weights are not equal.

Step-by-step solution

Given information(part a)

Given in the question that, A company packages apples by weight. One of the weight grades is Class A apples. Class A apples have a mean weight of $150g$, and there is a maximum allowed weight tolerance of$5\%$above or below the mean for apples in the same consumer package. A batch of apples is selected to be included in a Class A apple package.

Explanation(part a)

The null hypothesis is stated below:

$H_0$: The weights are equal.

Against the alternative hypothesis as shown below:

$H_a$: The weights are not equal.

The goodness-of-fit test can be conducted here because all entries are greater than or equal to five. Therefore, the degree of freedom can be calculated as shown below:

$df=$number of cells $-1$

$=15-1$

$=14$

Calculations(part a)

In excel, do the calculations shown below:

Hence, the test statistic is $16.56$.

p-value calculations(part a)

The$p-$value can be calculated in excel by using CHIDIST $()$formula as shown below:

Hence, the $p$-value is$0.280$.

i. Alpha: $0.05$

ii: Decision: Do not reject the null hypothesis $H_0$

iii. Reason for decision: Because $p-$value $>\alpha$

iv. Conclusion: There is insufficient evidence to insure that the weights are not equal.

Final answer(part a)

There is insufficient evidence to insure that the weights are not equal.

Given information(part b)

Given in the question that, A company packages apples by weight. One of the weight grades is Class $A$apples. Class$A$ apples have a mean weight of $150g$, and there is a maximum allowed weight tolerance of $5\%$above or below the mean for apples in the same consumer package. A batch of apples is selected to be included in a Class$A$ apple package.

Explanation(part b)

The null hypothesis is stated below:

$H_0$: The weights are equal.

Against the alternative hypothesis as shown below:

$H_a$: The weights are not equal.

Goodness-of-fit test can be conducted here because all entry are greater than or equal to five. Therefore, the degree of freedom can be calculated as shown below:

$df=$number of cells $-1$

$=15-1$

$=14$

Calculations (part b)

In excel, do the calculations shown below:

Hence, the test statistic is $16.56$.

Step 9: p-value calculation(part b)

The$p$-value can be calculated in excel by using CHIDIST ( ) formula as shown below:

Hence, the $p$-value is $0.280$.

i. Alpha:$0.01$

ii: Decision: Do not reject the null hypothesis$H_0$

iii. Reason for decision: Because $p-$value$>\alpha$

iv. Conclusion: There is insufficient evidence to insure that the weights are not equal.