You have just taken ownership of a pizza shop. previous owner told you that you would save money if you bought the mozzarella cheese in a 4.5 pound slab on the average. Each time you purchase a slab of cheese, you weigh it to ensure that you are receiving a standard deviation of 72 ounces of cheese. The results of 7 random measurements are 70, 69, 73, 68, 71, 69 The and 71 ounces. Are these differences due to chance or is the distributor giving you less cheese than you deserve? i. State the hypotheses. ii. Calculate the test statistic. iii. Would the null hypothesis be rejected at the 10% level? The 5% level? The 1% level?
Answers
Answer:
Back in the early 1900’s a chemist at a brewery in Ireland discovered that when he was working with very small samples, the distributions of the mean differed significantly from the normal distribution. He noticed that as his sample sizes changed, the shape of the distribution changed as well. He published his results under the pseudonym ‘Student’ and this concept and the distributions for small sample sizes are now known as “Student’s distributions.”
distributions are a family of distributions that, like the normal distribution, are symmetrical and bell-shaped and centered on a mean. However, the distribution shape changes as the sample size changes. Therefore, there is a specific shape or distribution for every sample of a given size (see figure below; each distribution has a different value of , the number of degrees of freedom, which is 1 less than the size of the sample).
We use the Student's distribution in hypothesis testing the same way that we use the normal distribution. Each row in the distribution table (see link below) represents a different distribution and each distribution is associated with a unique number of degrees of freedom (the number of observations minus one). The column headings in the table represent the portion of the area in the tails of the distribution – we use the numbers in the table just as we used the scores.
As the number of observations gets larger, the distribution approaches the shape of the normal distribution. In general, once the sample size is large enough - usually about 30 - we would use the normal distribution or the table instead. Note that usually in practice, if the standard deviation is known then the normal distribution is used regardless of the sample size.
In calculating the test statistic, we use the formula:
where:
is the test statistic and has degrees of freedom.
is the sample mean
is the population mean under the null hypothesis.
is the sample standard
Step-by-step explanation:
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You have just taken ownership of a pizza shop. The previous owner told you that you
would save money if you bought the mozzarella cheese in a 4.5 pound slab. Each time
you purchase a slab of cheese, you weigh it to ensure that you are receiving 72 ounces
of cheese. The results of 7 random measurements are 70, 69, 73, 68, 71, 69 and 71
ounces. Are these differences due to chance or is the distributor giving you less cheese
than you deserve?