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According to the latest financial reports from a sporting goods store, the mean sales per customer was $75\$75 with a population standard deviation of $6\$6. The store manager believes 3939 randomly selected customers spent more per transaction.\newlineUse a calculator to find the probability that the sample mean of sales per customer is between $76\$76 and $77\$77 dollars. Round to two decimal places.

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Q. According to the latest financial reports from a sporting goods store, the mean sales per customer was $75\$75 with a population standard deviation of $6\$6. The store manager believes 3939 randomly selected customers spent more per transaction.\newlineUse a calculator to find the probability that the sample mean of sales per customer is between $76\$76 and $77\$77 dollars. Round to two decimal places.
  1. Identify Given Values and Formula: Identify the given values and the formula to use.\newlineWe are given:\newline- Mean sales per customer μ\mu = $75\$75\newline- Population standard deviation σ\sigma = $6\$6\newline- Sample size nn = 3939\newline- We want to find the probability that the sample mean xˉ\bar{x} is between $76\$76 and $77\$77.\newlineWe will use the z-score formula for the sample mean:\newlinez=xˉ−μσ/nz = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}\newlineWe will calculate two z-scores, one for $76\$76 and one for $77\$77, and then use the standard normal distribution to find the probabilities.
  2. Calculate Z-Score for $76\$76: Calculate the z-score for $76\$76. Using the z-score formula: z=76−756/39z = \frac{76 - 75}{6 / \sqrt{39}} First, calculate the denominator σ/n\sigma / \sqrt{n}: σ/n=639≈66.245≈0.961\sigma / \sqrt{n} = \frac{6}{\sqrt{39}} \approx \frac{6}{6.245} \approx 0.961 Now, calculate the z-score: z=76−750.961≈10.961≈1.041z = \frac{76 - 75}{0.961} \approx \frac{1}{0.961} \approx 1.041
  3. Calculate Z-Score for $77\$77: Calculate the z-score for $77\$77. Using the z-score formula: z=77−756/39z = \frac{77 - 75}{6 / \sqrt{39}} We already calculated the denominator σ/n\sigma / \sqrt{n} in the previous step, so we can use it again: z=77−750.961≈20.961≈2.081z = \frac{77 - 75}{0.961} \approx \frac{2}{0.961} \approx 2.081
  4. Use Standard Normal Distribution: Use the standard normal distribution to find the probability for each zz-score.\newlineWe need to find the probability that zz is less than 1.0411.041 and then the probability that zz is less than 2.0812.081. The difference between these two probabilities will give us the probability that the sample mean is between $76\$76 and $77\$77.\newlineUsing a standard normal distribution table or calculator:\newlineP(z < 1.041) \approx 0.851\newlineP(z < 2.081) \approx 0.981
  5. Calculate Probability: Calculate the probability that the sample mean is between $76\$76 and $77\$77. We subtract the probability of zz being less than 1.0411.041 from the probability of zz being less than 2.0812.081: Probability = P(z < 2.081) - P(z < 1.041) Probability ≈0.981−0.851\approx 0.981 - 0.851 Probability ≈0.130\approx 0.130

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