Question

The following data comparing wait times at two rides at Disney are listed below: Position Pirates Splash Mountain Sample Size 32 30 Average Wait Time (In Minutes) 14.68 18.77 Population Standard Deviation 11.87 16.79 What is the 98% confidence interval for the difference in wait times between pirates and splash mountain? What is the test statistic for testing to see if there is a significant difference in wait times between pirates and splash mountain?

Answer 1

The 98% confidence interval for the difference in wait times between Pirates and Splash Mountain can be calculated using the formula that includes the sample means, sample sizes, and population standard deviations. To test for a significant difference, the two-sample z-test is used to find the z-score which is then compared to the critical z-value.

Step-by-step explanation:

To find the 98% confidence interval for the difference in wait times between Pirates and Splash Mountain, we assume the wait times are normally distributed and use the formula for the confidence interval of the difference between two means when we know the population standard deviations. The formula for a two-sample z-interval is:

CI = (μ1 - μ2) ± z* √((σ2^2/n1) + (σ2^2/n2))

Where:

• μ1 and μ2 are the sample means (14.68 and 18.77 respectively).
• n1 and n2 are the sample sizes (32 and 30 respectively).
• σ1 and σ2 are the population standard deviations (11.87 and 16.79 respectively).
• z* is the z-value corresponding to the desired confidence level (2.326 for 98%).

Using the above values, we can calculate the confidence interval.

To test for a significant difference in wait times, we use the formula for the test statistic in a two-sample z-test:

Z = (μ1 - μ2) / √((σ2^2/n1) + (σ2^2/n2))

Plugging in the values, we can calculate the z-score. If this z-score is greater than the critical value for our alpha level (2.326 for a 2% tail), we can conclude there is a significant difference in wait times.

Answer 2

a)
`(14.68 -18.77) - 2.39 \sqrt{(11.87^2)/(32)+(16.79^2)/(30)} =-12.968`

`(14.68 -18.77) + 2.39 \sqrt{(11.87^2)/(32)+(16.79^2)/(30)} =4.788`

b)
`t=\frac{14.68-18.77}{\sqrt{(11.87^2)/(32)+(16.79^2)/(30)}}}=-1.10`

Step-by-step explanation:

Data given and notation

`\bar X_(A)=14.68` represent the mean for Pirates

`\bar X_(B)=18.77` represent the mean for Splash Mountain

`s_(A)=11.87` represent the sample standard deviation for the sample Pirates

`s_(B)=16.79` represent the sample standard deviation for the sample Slpash Mountain

`n_(A)=32` sample size selected for Pirates

`n_(B)=30` sample size selected for Splash Mountain

`\alpha=0.02` represent the significance level for the hypothesis test.

t would represent the statistic (variable of interest)

`p_v` represent the p value for the test (variable of interest)

Part a

The confidence interval would be given by:

`(\bar X_A -\bar X_B) \pm t_(\alpha/2) \sqrt{(s^2_(A))/(n_(A))+(s^2_(B))/(n_(B))}`

The degrees of freedom are given by:

`df = n_A +n_B -2 = 32+30-2 = 60`

Since we want 98% of confidence the significance level is
`\alpha =1-0.98 =0.02` and
`\alpha/2 =0.01`, we can find in the t distribution with df =60 a critical value that accumulates 0.01 of the area on each tail and we got:

`t_(\alpha/2)= 2.39`

And replacing we got for the confidence interval:

`(14.68 -18.77) - 2.39 \sqrt{(11.87^2)/(32)+(16.79^2)/(30)} =-12.968`

`(14.68 -18.77) + 2.39 \sqrt{(11.87^2)/(32)+(16.79^2)/(30)} =4.788`

Part b

State the null and alternative hypotheses.

We need to conduct a hypothesis in order to check if the means are equal, the system of hypothesis would be:

Null hypothesis:
`\mu_(A) = \mu_(B)`

Alternative hypothesis:
`\mu_(A) \\eq \mu_(B)`

the statistic is given by:

`t=\frac{\bar X_(A)-\bar X_(B)}{\sqrt{(s^2_(A))/(n_(A))+(s^2_(B))/(n_(B))}}` (1)

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine whether the means of two groups are equal to each other".

Calculate the statistic

We can replace in formula (1) the info given like this:

`t=\frac{14.68-18.77}{\sqrt{(11.87^2)/(32)+(16.79^2)/(30)}}}=-1.10`