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love history [14]
3 years ago
6

An article includes the accompanying data on compression strength (lb) for a sample of 12-oz aluminum cans filled with strawberr

y drink and another sample filled with cola.Beverage Sample Sample Sample Size Mean SDStrawberry Drink 10 537 22Cola 10 559 17Does the data suggest that the extra carbonation of cola results in a higher average compression strength? Base your answer on a P-value.State the relevant hypotheses. (Use \mu1 for the strawberry drink and \mu2 for the cola.)H0: \mu1 - \mu2 = 0 Ha: \mu1 - \mu2 > 0H0: \mu1 - \mu2 = 0 Ha: \mu1 - \mu2 < 0 H0: \mu1 - \mu2 = 0 Ha: \mu1 - \mu2\geq 0H0: \mu1 - \mu2 = 0 Ha: \mu1 - \mu2\neq 0Compute the test statistic value and find the P-value. (Round your test statistic to three decimal places and your P-value to four decimal places.)t = ___P-value = ___State the conclusion in the problem context. (Use \alpha = 0.05.)Reject H0. The data suggests that cola has a higher average compression strength than the strawberry drink.Reject H0. The data does not suggest that cola has a higher average compression strength than the strawberry drink. Fail to reject H0. The data suggests that cola has a higher average compression strength than the strawberry drink.Fail to reject H0. The data does not suggest that cola has a higher average compression strength than the strawberry drink.What assumptions are necessary for your analysis?The distributions of compression strengths are approximately normal.The distributions of compression strengths have equal means. The distributions of compression strengths are the same.The distributions of compression strengths have equal variances.

Mathematics
1 answer:
Sliva [168]3 years ago
7 0

Answer:

A) Option B is correct.

H₀: μ₁ = μ₂

Hₐ: μ₁ - μ₂ < 0

B) t = -2.502

p-value = 0.0112

C) Option A is correct.

Reject H₀. The data suggests that cola has a higher average compression strength than the strawberry drink.

D) Option A is correct.

The distributions of compression strengths are approximately normal.

Step-by-step explanation:

The complete Question is presented in the two attached images to this answer.

A) To perform this test we first define the null and alternative hypothesis.

The null hypothesis plays the devil's advocate and usually takes the form of the opposite of the theory to be tested. It usually contains the signs =, ≤ and ≥ depending on the directions of the test.

While, the alternative hypothesis usually confirms the the theory being tested by the experimental setup. It usually contains the signs ≠, < and > depending on the directions of the test.

For this question, we want to test if the extra carbonation of cola results in a higher average compression strength. That is, that cola has a higher average compression strength than the strawberry drink.

Hence, the null hypothesis would be that there isn't significant evidence to suggest that the extra carbonation of cola results in a higher average compression strength, that is, cola has a higher average compression strength than the strawberry drink.

The alternative hypothesis is that there is significant evidence to suggest that the extra carbonation of cola results in a higher average compression strength, that is, cola has a higher average compression strength than the strawberry drink.

Mathematically, if the average compression strength of strawberry drink is μ₁, the average compression strength of cola is μ₂ and the difference in compression strengths is μ = μ₁ - μ₂

The null hypothesis is represented as

H₀: μ = 0 or μ₁ = μ₂

The alternative hypothesis is represented as

Hₐ: μ < 0 or μ₁ - μ₂ < 0

B) So, to perform this test, we need to compute the test statistic

Test statistic for 2 sample mean data is given as

Test statistic = (μ₁ - μ₂))/σ

σ = √[(s₂²/n₂) + (s₁²/n₁)]

μ₁ = average compression strength of strawberry drink = 537

n₁ = sample size of the sample of strawberry drink in cans surveyed = 10

s₁ = standard deviation of the compression strength of strawberry drink in cans surveyed= 22

μ₂ = average compression strength of cola = 559

n₂ = sample size of the sample of cola in cans surveyed = 10

s₂ = standard deviation of the compression strength of strawberry drink in cans surveyed = 17

σ = [(17²/10) + (22²/10)] = 77.5903160379 = 8.792

We will use the t-distribution as no information on population standard deviation is provided

t = (537 - 559) ÷ 8.792

= -2.502 = -2.50

checking the tables for the p-value of this t-statistic

Degree of freedom = df = n₁ + n₂ - 2 = 10 + 10- 2 = 18

Significance level = 0.05

The hypothesis test uses a one-tailed condition because we're testing in only one direction (whether compression strength of cola in can is greater).

p-value (for t = -2.50, at 0.05 significance level, df = 18, with a one tailed condition) = 0.011154 = 0.0112 to 4 d.p.

C) The interpretation of p-values is that

When the (p-value > significance level), we fail to reject the null hypothesis and when the (p-value < significance level), we reject the null hypothesis and accept the alternative hypothesis.

So, for this question, significance level = 0.05

p-value = 0.0112

0.0112 < 0.05

Hence,

p-value < significance level

This means that we reject the null hypothesis accept the alternative hypothesis & say that the extra carbonation of cola results in a higher average compression strength, that is, cola has a higher average compression strength than the strawberry drink.

D) The necessary conditions required before a t-test is deemed valid include.

- The samples used must be a random sample of the population distribution with each variable in the sample independent of other one.

- The distribution of the population where the samples were extracted from must be normal or approximately normal to ensure some degree of normality for the samples.

Hence, the necessary assumption for this t-test among the options is that the distributions of compression strengths are approximately normal.

Hope this Helps!!!

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Choice 4 is false. Following the steps mentioned here will create an altitude line (see choice 1)

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Tems11 [23]

<em>Question:</em>

<em>If one basket contains five bottles each weighing 7 1/4 and the other basket has six apples each weighing 6 1/2 ounces. Which basket is heavier and by how much ?</em>

Answer:

The basket of bottles weighs more than the basket of apples by 3\frac{3}{4} ounces

Step-by-step explanation:

Given

Bottles = 5

Weight(Bottle) = 7\frac{1}{4}

Apples = 6

Weight(Apple) = 6\frac{1}{2}

Required

Determine which basket is heavy

and by how much?

First, we need the total weight of each basket.

For the bottles' basket

Total\ Weight = Weight * Bottles

Total\ Weight = 7\frac{1}{4} * 5

Total\ Weight = \frac{29}{4} * 5

Total\ Weight = \frac{29 * 5}{4}

Total\ Weight = \frac{145}{4}

Total\ Weight = 36\frac{1}{4}

For the bottles' basket

Total\ Weight = Weight * Apples

Total\ Weight = 6\frac{1}{2} * 5

Total\ Weight = \frac{13}{2} * 5

Total\ Weight = \frac{13* 5}{2}

Total\ Weight = \frac{65}{2}

Total\ Weight = 32\frac{1}{2}

By comparing both total weights, 36\frac{1}{4} > 32\frac{1}{2}

Hence:

The basket of bottles weighs more than the basket of apples

Calculating the difference:

Difference = 36\frac{1}{4} - 32\frac{1}{2}

Difference = \frac{145}{4} - \frac{65}{2}

Difference = \frac{145 - 130}{4}

Difference = \frac{15}{4}

Difference = 3\frac{3}{4}

Hence:

It weighs more by 3\frac{3}{4} ounces

8 0
3 years ago
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Answer:

Number of units of first product = 112

Number of units of second product = 400

Step-by-step explanation:

Given that there are two products which are to be shipped.

<em>First product details:</em>

Unit volume = 10 cu ft

Weight of one unit = 50 lb

<em>Second product details:</em>

Unit volume = 3 cu ft

Weight of one unit = 40 lb

Total volume that can be carried by a truck = 2320 cu ft

Total weight that can be carried by a truck = 21600 cu ft

To find:

The number of units of each product that can be transported in one shipment in the truck?

Solution:

Let number of units of first product = x

Let number of units of second product = y

We can write down two linear equations in two variables here and solving them will give us the answer we are required to find.

As per question statement, the equations can be written as:

10x+3y = 2320 ...... (1)\\50x+40y = 21600\\\Rightarrow  5x+4y = 2160...... (2)

Multiplying equation (2) with 2 and subtracting equation (1) from it:

8y - 3y = 4320-2320\\\Rightarrow 5y = 2000\\\Rightarrow \bold{ y = 400}

By equation (1):

10x+1200 = 2320\\\Rightarrow 10x = 1120\\\Rightarrow \bold{x = 112}

Answer is:

Number of units of first product = 112

Number of units of second product = 400

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