Mathematics, 14.07.2020 01:01, Puppy9118

McDowell's is famous for their Big Mick. As their most ordered item – the Big Mick grew in popularity, so did their small corner store locations. The secret to their success is the secret ingredient used in the Big Mick. To ensure that a corner store is maintaining the same quality and taste as any of their main stores by using the secret ingredient properly, they collected some samples and measured the percentage (proportion) of the secret ingredient used in the Big Micks. The summary data is presented below: Mean Store Location Sample Size Standard Deviation
Corner 53 0.035 0.340
Main 51 0.960 1.046
(a) Conduct a hypothesis test with 90% reliability to see if the mean secret ingredient percentage in Big Micks from the two stores are equal.
(b) Compute a confidence interval with a = 0.10 and explain why the result is identical to part (a)

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a) At a significance level of 0.1, there is not enough evidence to support the claim that the mean secret ingredient percentage in Big Micks from the two stores are not equal.

b) The 90% confidence interval for the difference between means is (-0.632, 0.022).

As the value 0 is included as a probable value for the difference of means, we can conclude that there is no enough evidence that the difference of means is significantly different from 0.

Both the hypothesis test and the confidence interval have the same result, as both use the same sample and assumptions.

Step-by-step explanation:

a) This is a hypothesis test for the difference between populations means.

The claim is that the mean secret ingredient percentage in Big Micks from the two stores are not equal.

Then, the null and alternative hypothesis are:

$H_0: \mu_1-\mu_2=0\\\\H_a:\mu_1-\mu_2\neq 0$

The significance level is 0.1.

The sample 1 (corner stores), of size n1=53 has a mean of 0.035 and a standard deviation of 1.046.

The sample 2 (main stores), of size n2=51 has a mean of 0.34 and a standard deviation of 0.96.

The difference between sample means is Md=-0.305.

M_d=M_1-M_2=0.035-0.34=-0.305

The estimated standard error of the difference between means is computed using the formula:

$s_{M_d}=\sqrt{\dfrac{\sigma_1^2}{n_1}+\dfrac{\sigma_2^2}{n_2}}=\sqrt{\dfrac{1.046^2}{53}+\dfrac{0.96^2}{51}}\\\\\\s_{M_d}=\sqrt{0.021+0.018}=\sqrt{0.039}=0.1968$

Then, we can calculate the t-statistic as:

$t=\dfrac{M_d-(\mu_1-\mu_2)}{s_{M_d}}=\dfrac{-0.305-0}{0.1968}=\dfrac{-0.305}{0.1968}=-1.55$

The degrees of freedom for this test are:

df=n_1+n_2-2=53+51-2=102

This test is a two-tailed test, with 102 degrees of freedom and t=-1.55, so the P-value for this test is calculated as (using a t-table):

$\text{P-value}=2\cdot P(t$

As the P-value (0.12) is greater than the significance level (0.1), the effect is not significant.

The null hypothesis failed to be rejected.

At a significance level of 0.1, there is not enough evidence to support the claim that the mean secret ingredient percentage in Big Micks from the two stores are not equal.

b) We have to calculate a 90% confidence interval for the difference between means.

The critical t-value for a 90% confidence interval is t=1.66.

The margin of error (MOE) can be calculated as:

$MOE=t\cdot s_{M_d}=1.66 \cdot 0.1968=0.327$

Then, the lower and upper bounds of the confidence interval are:

$LL=M_d-t \cdot s_{M_d} = -0.305-0.327=-0.632\\\\UL=M_d+t \cdot s_{M_d} = -0.305+0.327=0.022$

The 90% confidence interval for the difference between means is (-0.632, 0.022).

As the value 0 is included as a probable value for the difference of means, we can conclude that there is no enough evidence that the difference of means is significantly different from 0.

answered: Guest

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answered: Guest

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McDowell's is famous for their Big Mick. As their most ordered item – the Big Mick grew in popularit...